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WASHINGTON OBSERVATIONS FOR 1885.— APPENDIX III

THE

SOLAR PARALLAX

AND ns

RELATED CONSTANTS,

INCLUDING THE

FIGURE AND DENSITY OF THE EARTH.

BY

WM. HARKNESS,

PROFESSOR OF MATHEMATICS, U. S. NAVY.

DEPARTMENT OF PHYSICS

Case School of BpplteD science

CLEVELAND OH I©

WASHINGTON:

GOVERNMENT PRINTING OFFICE.

i 8 9 I .

TABLE OF CONTENTS

Tag*.

1. Introductory 1

2. Algebraic notation, and citation of authorities 1

3. Relations between units of length 3

4. Size and figure of the Earth 3

5. Length of the seconds pendulum 6

6. Length of the year 10

7. The eccentricity of the Earth's orbit, and the constant (1+x) 11

8. The length of the month 12

9. The constants//, m, (1-\-h'), e2, y and I, pertaining to the Moon 10

10. Observed value of the parallactic inequality of the Moon 18

11. Observed value of the lunar inequality of the Earth 19

12. Observed value of the lunar parallax 20

13. The constant of precession 22

14. The constant of nutation 25

15. The constant of aberration 25

16. The light equation 28

17. V, the velocity of light in vacuo 29

18. Masses of the planets 33

19. Trigonometrical determinations of the solar parallax 51

20. General forms of the conditional equations 56

21. The least square adjustment 64

22. Numerical values of the corrections by adjustment 6S

23. Additional formula? for precession 73

24. The density, flattening, and moments of inertia of the Earth 89

25. Uncertainty in the value of £, and its effect upon the other constants 102

26. Mass of the Moon from observations of the tides 112

27. A more comprehensive least square adjustment 121

28. The electric constant, v 135

29. Supplementary data 136

30. Summary of results 138

31. Concluding remarks 142

32. Bibliography 146

Index 165

m

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

1.— INTRODUCTORY.

Hitherto it has been customary to endeavor to determine the solar parallax as if
it were an independent constant, and the result is a mass of discordant values, all of
which are more or less affected by constant errors, and none of which command any-
thing like universal assent. But, in truth, the solar parallax is not an independent
constant. On the contrary, it is entangled with the lunar parallax, the constants of
precession and nutation, the parallactic inequality of the Moon, the lunar inequality
of the Earth, the masses of the Earth and Moon, the ratio of the solar and lunar tides,
the constant of aberration, the velocity of light, and the light equation ; and according
to the most elementary mathematical principles, it should be determined simultane-
ously with all these quantities, by means of a least square adjustment. No other
method offers anything like so much promise of eliminating the ever present constant
errors, and therefore an attempt will be made to develop it here. The equations
connecting the quantities mentioned are known, but for the sake of completeness
their derivation will be given. The theory of these equations and the discussion
of the numerical quantities which they involve are entirely distinct subjects, and as
clearness will be gained by separating them, we will begin by investigating the numeri-
cal values, both of the constants, and of the quantities to be adjusted.

2.— ALGEBRAIC NOTATION, AND CITATION OF AUTHORITIES.

Except where otherwise stated, the following notation will be employed in alge-
braic formula? :

a zz equatorial semi-axis of the globe of the Earth, if that body is regarded as a
spheroid ; or major equatorial semi-axis, if it is regarded as an ellipsoid.

a' •=. minor equatorial semi-axis of the Earth, when that body is regarded as an
ellipsoid.

b — polar semi-axis of the Earth.

ax = that distance between the Earth and the Sun which would satisfy Kepler's
third law.

* In its original form this paper was a private investigation, and as such was read before the Philosophical Society of "\\ ash-
ington on October 13, 1888. Since then it has been taken up in connection with my official work, to which it is closely related,
and many important details have been added. — W. H.

6987 1

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

a2 zz that distance between the Earth and the Moon which would satisfy Kepler's

third law.
E zz the combined mass of the Earth and Moon.
E' zz the mass of the Earth, excluding- the Moon.
e zz eccentricity of the globe of the Earth.
<?! zz eccentricity of the Earth's orbit.
e2 zz eccentricity of the Moon's orbit,

g zz observed force of gravity at a specified point upon the Earth's surface.
I zz inclination of the Moon's orbit to the plane of the ecliptic.
k zz Gauss's constant for the solar system.
L zz constant of the Earth's lunar inequality.
I zz length of a simple pendulum vibrating in the time t.
M zz mass of the Moon.

m zz ratio of the mean motions of the Sun and Moon.
£1 zz the constant of nutation.
|£ zz the constant of luni-solar precession.
P zz the constant of lunar parallax.
p zz the constant of solar parallax.
Q zz the parallactic inequality of the Moon.
r zz that value of the mean distance from the Earth to the Sun which is adopted

in the solar tables.
t\ zz that value of the mean distance from the Earth to the Moon which is

adopted in the lunar tables.

5 zz mass of the Sun.

s' zz geocentric latitude of the Moon.

T zz length of the sidereal year, expressed in seconds of mean time.

Tx zz length of the sidereal month, expressed in seconds of mean time.

t zz time.

tx zz number of mean solar seconds in a sidereal day.

V zz the velocity of light per second of mean time.

a zz the constant of aberration.

y zz Delauxay's constant, which is sin \ (inclination of lunar orbit to plane of

ecliptic).
* = (a — b)/a zz the quantity variously designated as the ellipticity, compression,

or flattening' of the Earth.

6 zz the time taken by light to traverse the mean radius of the Earth's orbit.
x zza constant, such that a^zzr (i + x)

x' zz a constant, such that a2 = >\ (i + x')

M = regression of Moon's node, relatively to the line of equinoxes, in 365^ days.
v zz the heliocentric longitude of the Earth.
v' zz the geocentric longitude of the Moon.

p zza factor, varying with the latitude, such that the radius of the globe of the
Earth at latitude <p is ap.
zz centrifugal force at latitude cp, divided by gravity at the same latitude.

a

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. 3

<p — geographic latitude.

q>' zz geocentric latitude.

ip zz the luni-solar precession.

ip! zz the general precession.

co0 — the mean obliquity of the fixed ecliptic at the initial epoch.

go — the obliquity of the fixed ecliptic at the time t.

&>! zz the obliquity of the moving ecliptic at the time t.

For a list of the principal works consulted in the preparation of the present paper,
and an explanation of the method by which they are cited in the foot-notes, the reader
is referred to the section on bibliography, pages 146-165.

3.— RELATIONS BETWEEN UNITS OF LENGTH.

For interchanging the various units of length employed by different authors, the
following ratios, based upon Gen. A. R. Clarke's determinations,* will be used
throughout this paper :

Logarithms.

i meter — 3*280 869 33 feet 0*515988934

1 kilometer rzO"62i 376 767 mile 9793 355 OI1

1 toise zz 6-394 533 48 feet 0*805 808 865

1 toise zz 1*949036318 meters 0*289819931

1 Paris line zz 0*002 255 829072 meter 7*353 306 189

1 statute mile zz 1*609 329 561 kilometers 0*206 644 989

1 English inch = 0*025 399 772 meter 8*404 829 820

4.— SIZE AND FIGURE OF THE EARTH.

When the Earth is regarded as a spheroid, let a be its equatorial and b its polar
semi-axis; and when it is regarded as an ellipsoid, let a and a' be respectively its major
and minor equatorial, and b its polar semi-axis. Then the flattening will be given by

the formula

a — b , x

£=— — (0

a

and the ratio of the axes will be

a:b=l:--i (2)

e e

In 1 799 Mlchain and Delambre found from a combination of the arc of 90 40'
which they had measured between Dunkerque and Barcelona with the arc of 30 07'
measured in Peru by Bouguer and La CoNDAMiNEf

Log. a (in meters) zz 6*804 53°5 °74
Log. b (in meters) zz 6*803 22&2 744
Whence

a — 6 375 738*66 meters zz 20917 965 feet

b zz 6 356 649*63 meters zz 20 855 ^37 feet
frz 1/334

*'3. P- '57, and 23, p. 280. f2S. T- 3. PP- >96 and 432-

4 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

In 1830, from fourteen meridional arcs having- an amplitude of 590 29', and four
arcs of parallel having an amplitude of 220 41', Sir (then Professor) George B. Airy
found *

a=. 20923 713 feet

b — 20853 810 feet

£— 1/299-3249

In 1 84 1, from ten meridional arcs having a total amplitude of 500 36', to which
he applied a more rigorous analysis than ha,d before been used, Bessel found f

a = 3 272 077' 14 toises = 20923 407 feet
b =. 3 261 139*33 toises =r 20 853 465 feet

€= 1/299-1529

In 1858, from a discussion of eight meridional arcs having a total amplitude of
670 08', General (then Captain) A. R. Clarke found, when the curvature of the
meridians was not restricted to an elliptic form J

a — 20927 197 ±385 feet
& = 20 855 493 ±257 feet

£= 1/291-8554

And when the curvature was restricted to an elliptic form§

a =z 20 926 348 rb 1 86 feet
£ = 20855 233 ±239 feet
£ = 1/294 2607

In 1859 General T. F. de Schubert advanced the idea that the well-known dis-
cordances between the astronomical and geodetic differences of latitude and longitude
of points upon the Earth's surface arise mainly from the assumption that the Earth is
a spheroid, when in truth it is an ellipsoid ; and, in accordance with that hypothesis,
he found from eight meridional arcs having a total amplitude of 72 ° 37' ||

a = 3 272 671-5 toises = 20927 207 feet
a = 3 272 303-2 toises rz 20924852 feet
3 261 467-9 toises = 20 855 566 feet

6 =

In i860 General A. R. Clarke repeated General de Schubert's investigation by
applying a much more exact analysis to five meridional arcs having a total amplitude
of 760 35'. He founds

a=z 20 926485 feet
a' = 20921 177 feet

6=20853 768 d= 953 feet

•17, p. 220. t'9. pn6. t 2i, p. 765. §21, p. 771. || 26, p. 31. Tf 22, p. 39.

ON THE SOLAR PARAL1 \\ AND ITS RELATED CONSTANTS. c

In 1866 the comparisons made at Southampton showed that the hitherto accepted
relations of the principal standards of length were slightly erroneous, and to correct
the error thence arising- General Clarke recomputed the axes last given, and found*

a zz 20926 350 feet
a zz 20919 972 feet
b zz 20853 429 feet

By modifying his equations so as to make them represent a spheroid, he found
from the same dataf

a zz 20 926 062 feet
&zz 20 855 121 feet
e= 1/294-9784

In 1878 the serious uncertainty respecting the unit of length employed by Colonel
Lambton in the measurement of the southern half of the Indian meridional arc had
been remedied by a complete remeasurement of that part of the triangulation ; the
latitudes of many stations in it had been determined ; the length of the arc had been
increased from 21 ° 21' to 230 50'; and an arc of longitude extending through io° 28'
had been measured. The data available for determining the size and figure of the
Earth were then the Russian arc of 250 20', the Anglo-French arc of 220 io', the
Indian meridional arc of 230 50', the Indian longitudinal arc of io° 28', the Cape arc
of 40 37', and the Peruvian arc of 30 07'. From these six arcs, having a total ampli-
tude of 890 32', General (then Colonel) Clarke found J

azz 20 926 629 feet

a! zz 20 925 105 feet

6zz 20 854 477 feet

In 1880, after considering the ellipsoidal theory, and calling special attention to
the fact that sufficient data are not yet available for fixing definitively the form of the
Earth, General Clarke reverted to the theory of a spheroid, and found from the arcs
he had employed in 1 878 §

a z= 20 926 202 feet & zz: 20 854 895 feet (3)

These last values will be adopted in the present paper. From them we have

(4)

a — b _

a 293-4663
a2-b2

e2 — - zz 0-006803 481 019 (5)

a1

72

tan cp' — , tan cp — ( 1 — e2) tan cp — 0*993 I96 5 1 9 tan cp (6)

a2

_ 1 — e2 (2 — e-) sin2 g>_ cos cp / x

,a

P — —

1 — c- sin2 cp cos cp' cos {cp — cp')

*23, p. 285. U3, P-287. {24, p. 92. ? 13, p. 319-

6 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

And with sin cp z= \/&

<p = 35° 15' 5179" <p' = 35° 04' 4876" p2= 0-997 742 482 (8)

5.— LENGTH OF THE SECONDS PENDULUM.

Let I be the length of the simple pendulum which beats seconds of mean solar
time when vibrating- through an infinitely small arc, in vacuo, at the level of the sea,
in latitude cp. For determining the value of / the following data are available :

In 1 799, from observations made at 1 5 stations, whose latitudes ranged from
+ 670 05' to — 330 56', the whole combined in such a way as to make the absolute
length of the pendulum depend solely upon Borda's measurements at the Paris Ob-
servatory, La Place found*

£ = 0739 502 +0-004 2°8 sin2 cp meter

But this is the length of the decimal pendulum making 100 000 vibrations in a
mean solar day. To reduce it to the sexagesimal pendulum making 86 400 vibrations,
it must be multiplied by

1000
"8647

\ — i'339 59i 906

and then it becomes

£ = 0-990631 +0-005 637 sin2 cp meter

In 1 8 1 6, from observations made at 31 stations, whose latitudes ranged from
+ 740 53' to — 510 21', Mathieu found, for the decimal pendulum,!

/ zr 0-739 586 + 0004 080 6 sin2 cp meter
which becomes for the sexagesimal pendulum

I — 0990 743 + 0005 466 sin2 cp meter

In 1 82 1, from observations at 8 stations, situated in France and Scotland, between
latitudes + 380 40' and + 6o° 45', Biot found, for the decimal pendulum,!

I =z 0739 687 686 + 0-003 986 392 sin2 cp meter
which becomes for the sexagesimal pendulum

I zz 0*990 880 + 0-005 34° sm2 <P meter

In 1825, from his own observations at 13 stations having latitudes ranging from
+ 79° 5°' to — 120 59', Captain (afterwards General Sir) Edward Sabine found §

/z= 39 01 5 68 +0-20213 sin2 cp English inches
= 0990 989 + 0-005 T34 sin2 cp meter

*7, T. 2, livre3, chap. S.sec. 42. f32, p. 332. {20, p. 575. ?35>P-352.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. J

And from a combination of his own stations, those of the British survey, and those
of the French arc; in all 25 stations, having latitudes ranging from -f- 79° 50' to
— 12° 59', he found

/ =139-0 15 20 + 0202 45 sin2 cp inches
zr 0-990977 +0*005 l42 sin2 <P meter

In 1827, from observations made at 41 stations, whose latitudes ranged from
-f 790 50' to — 5 1° 35', M. Saigey found*

I =z 099 1026 + 0005 072 sin2 cp meter

In 1829, from observations made at 5 stations, whose latitudes ranged from o° to

+ 67° 04', PoNTECOULANT fouildf

1 = 0*990 555 + 0-005 679 sin2 cp meter

In 1830, from observations made at 49 stations, whose latitudes ranged from
+ 79° 50' to — 51 ° 35', Sir (then Professor) George B. Airy foundj

7-r: 39-oi6 77 + 0-20027 sin2 cp inches
r= 0991 017 + 0*005 087 sin2 cp meter

In 1833, by a process which is not clearly described, but wfhich seems to have
consisted in making the absolute length of the pendulum depend upon the experiments
of Borda at the Paris Observatory, while the coefficient of cos 2cp was made to depend
upon the compression of the Earth, assumed to be 1/288, Poisson found §

Z = o*993 512 (1 — 0002 588 cos 2cp) meter

To pass from the form

l — a(\ — b cos 2<p)
to the form

l—ot + fi sin2 cp
we have

a = a(i— b) fi—2ab

and therefore Poisson's formula is equivalent to

1 = 0-99094 1 + 0-005 142 sin2 cp meter

In 1869, from observations made at 51 stations, whose latitudes ranged from
+ 79° 5°' to — 5X° 35') Unferdinger found ||

Log. Z(in Paris lines) = 2-642 756 8 + [7-351 47] sin2 cp + [5-3 19 8] sin4 cp

+ [3'457] sin6 q>

*38.P-36- t '°» T. 2, p. 466. J 17, p. 230. ^15, T. i,p. 367; 34. PP- 32-33. and l6> T- 2'P-464- ||39»P-3I6-

8

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

In order to reduce this to the standard form, let us assume

I — 439-2923 + x + (2-2940 + y) sin2 <p

(9)

and then, by making- q> successively equal to o°, 200, 400, 6o°, and 8o°, and compar-
ing- the resulting values of / with those given by Unferdinger's expression, we shall
obtain the observation equations

o zz x + o-oooo y — 32
ozza; + o-i 170 — 10
o zz x : + 0-4 1 3 2 +12
ozz.r + 07500 —22
o zz x + 0-9698 — 78

where the absolute terms are in units of the fourth decimal place. The resulting
normal equations are

o zz + 5-0000 x + 2-2500 y — 1 30-0000

ozz+ 2-2500 +1-6874 — 88*3560
Whence

rrzz + 6-093 057 2/ = + 44'23?654

and by substitution in (9)

Izz 439-292 9 + 2-29842 sin2 <p Paris lines
zz 0-990970 + 0005 185 sin2 cp meter

In 1884, from observations made at 123 stations, whose latitudes ranged from
+ 790 50' to — 620 56', Helmert found*

I zz 0-990 9 1 8 [ 1 + (0:005 3 l ° zh H) sm2 0>] meter
zz 0-990 918 + (0*005 262 + 14) sin2 cp meter

For convenience of reference the preceding results are collected in Table I. In
the case of Unferdinger's formula, the value of I for latitude 45 ° was computed from
his original logarithmic expression, and not from that in the third column of the table.

Table I. — Formula' for the Length of the Seconds Pendulum.

Date.

Author.

/ in meters.

/(or <p = 45°

1799
1816
1821
1825
1827
1829
1830

1833
1869
1884

La Place .

Mathieu . .
Biot . . .
Sabine . .
Saigey . .
Pontecoulant
Airy . . .
Poisson . .
Unferdinger
Helmert . .

0990 631 -)- 0005 637 sin- (p
•990 743 5 466
•990 880 5 340
•990977 5142
•991 026 5 072

■990 555 5679
•991 017 5087

■990 941 5 J42
•99O97O 5185

0990 918 + 0-005 2°2 sin2 <p

o-993 45°
476

550
548
562

395
560
512
554
o-993 549

14, II Teil, p. 241.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTAT 9

A glance at the table shows that among the various authorities the range in the
length of the pendulum is nearly three times greater at the equator than al Latitude 450 ;
and that was to be expected, because most of the observations have been made within
the temperate zone. Judging from the data upon which they are based, the formulae
of Sabine, Saigey, Airy, Unferdinger, and Helmert seem the most trustworthy, and
their arithmetical mean gives

/ = o'993 5546 meter at lat. 450 (10)

which is probably not in error by so much as 00 r of a millimeter.

General Clarke's proof that the ellipticity of the Earth resulting from pendulum
observations does not differ sensibly from that obtained from terrestrial measurements,*
would be more satisfactory if he had used a more exact value of the ratio of the force
of gravity to the Earth's centrifugal force, but as his contention seems likely to prove
correct in the end, we will preserve the consistency of our data by deriving the coeffi-
cient of sin2 tp through Clairaut's theorem. The theoretical expression connecting
the length of the pendulum in any latitude with the size and figure of the Earth isf

, 7 ( . /iort a — b\ . o } , n

l=k\'+(M-—rn<p\ (,,)

where 1Q is the length of the pendulum at the equator, and tx is the number of mean
solar seconds in a sidereal day. Whence, by putting

I — /(, + B sin2 <p

we have

B_ 10a 10(a — b) £I2^

ti a

With ^=3-251 169 feet; ^ = 86 i64'io9, from (17); and the values of a and b
marked (3); formula (12) gives

B = coi 7 108 feet = 0^005 2144 meter

and by the combination of that result with (10), we obtain definitively

I — 0-990 947 -f-0005 214 sin2 (p meter / n

= 3-251 169 -fo'oi 7 108 sin2 cp feet

With sin2 q> ■=. yi, formula (13) gives

I — 3-256 872 English feet. (14)

*i3(P. 348. t">>T-2.P-428.

IO

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

6— LENGTH OF THE YEAR.

If we put m for the quantity by which the Sun's mean sidereal motion in 365^
days exceeds 3600, then the length of the sidereal year will be given by the expression

T = ^{i--^ + (-^)-ete|days

(15)

3600 ' V36C

According to Hansen and Olufsen,* m = — 22-56009"; whence, by formula (15)

T — 365-256 358 192 days — 365* o6h 09™ 09-347 88

According to Le Verrier the mean sidereal motion of the Sun in 365^ days isf
1 295 977-382 34" + 00603". Whence, mzz— 22 557 36"; and by formula (15)

T — 365-256357422 days=:365d o6h 09111 09-281 28

As the perturbations of the Earth are not taken account of in precisely the same
way by Hansen and Olufsen and by Le Verrier, the resulting values of the Sun's
mean sidereal motion given by these two authorities are not rigorously comparable,
but the systematic difference is very small. Neglecting it, we take the mean of the
two values of T just found, and thus obtain for the length of the sidereal year, expressed
in mean solar time

T = 365*2563578i days — 365* o6h 09m 093 14":= 31 558 1493 H seconds (16)

And the number of mean solar seconds in a sidereal day is

If we put

4 = 86 400* X llVAlllll = 86 1 64 099 65e
366 2^63578

m-\-rp1zza-\-bt

(17)

(18)

where tpx is the secular part of the general precession, the expression for the length of
the tropical year will be

-(^-(W?)M+etc-

T' = 365Ja

(19)

Taking for m the mean of the values given respectively by Hansen and Olufsen,
and by Le Verrier, namely,

m — — £ (22-560 09"+ 22-557 36") = — 22-558 72"

Uo, p. 1.

t4'» PP. 52 and 98.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. II

and putting m

lf>l = + 50*236 2 2/'-r-0,02 2 044"( -— '- *-)

V 100 J
we have

m+^1=4- 27-677 50" 4- 002 2 044 Y* ~ 1 85°)
Whence, by formula (19),

*

T = 365-242 1 99 853d - 0000 006 2 1 2 4^~l85°A
= 365d o5h 43m 46-067'- 0-536 75'(*~^5°)

(20)

The variation in the length of the sidereal or tropical year produced by a small
change in the adopted value of m, formula (15), or a, formula (19), may be readily
found from one or other of the expressions

dT = — 24-3 5<fm
dT = - 24- ssda ^2J)

where dT and dT' are in seconds of time, and dm and da in seconds of arc.

7.— THE ECCENTRICITY OF THE EARTH'S ORBIT AND THE CONSTANT (1 + «).

In their Tables du Soleil, published in 1853, Hansen and Olufsen gave for the
epoch 1850-0,* ex = 0016 771 20: but in his Darlegung, published in 1862, Hansen gave
for the epoch i8ooo,t ^ = 0016 792 26 Using the co-efficients of (23) to bring the
latter result up to the epoch 18500, it becomes

^rr 0016 771 035 (22)

. In 1858 Le Verrier gave J

ex = 3 459-28"- 0-087 55" (t- 1850)

-0000 282'f- l85°Y
V 100 J

or, in parts of radius

ex = 0016 771 063 — 0000000424 5 (t — 1850)

^_i85o\2 (23)

, n— 1050V

— o-ooooooooi 3671 =- )

V 100 J

Adopting the mean of (22) and (23), we have for the epoch 1850.0

e! = o-oi6 771 049 (24)

♦

40, p. 1. tSS.P>76- }4i,p. 102.

, 2 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

According to Le Verrier* .

and as we have put

it follows that

r — «! ( i + 2 ")

o1 = r (i + x)

3W

Here « is the Earth's mean sidereal motion, and a is a quantity depending upon
the perturbations of that motion by the other planets. For the numerical values of
these constants Le Verrier found f

Whence

Log. n zz6i 12 60 <T=:+ 2*506 7'

H = i'oooooi 289 5

and

1 + ^0-999998710 (25)

8.— LENGTH OF THE MONTH.

The best authorities for the data from which to derive the length of the month
are Airy, Hansen, and Newcomb.

The following notation will be employed :

g zzmean anomaly of the Moon.

g' zz mean anomaly of the Sun.

co zz distance of the lunar perigee from- the Moon's ascending node.

00 zz distance of the solar perigee from the Moon's ascending node.

0 zz supplement of the longitude of the Moon's ascending node.

tf>i zz the secular part of the general precession.

X zz sidereal movement of the solar perigee in 365^ days.
n" zz the Moon's mean sidereal motion in 365^ days.
m," zzthe Moon's mean synodical motion in 365^ days.
n2" zzthe Moon's mean tropical motion in 3651 days.
w3// zzthe Moon's mean nodical motion in 365^ days.
n" zzthe Moon's mean anomalistic motion in 365^ days.

*8, T. 2, pp. 30, 31, and 60. t8»T- 2> P- 59. and 41. P- "■

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

13

The mean motion of the Moon used by Damolskai; can best be found by taking
from his Table II* the mean longitudes for the years — 300 and — 2300 and dividing
their difference by 2000. The resulting mean tropical motion in 365^ days is

5347-420869 125 centesimal degrees

or, expressed in sexagesimal seconds, of which there are 3240 in each centesimal
degree, 17325 643-616 o". To that motion Airy found the correction -j- 0-596";! and
as he used Bessel's value of the precession, which for the epoch i8oo-o is 50-2235",
his value of the mean sidereal motion of the Moon in 365^ davs is

17325643-6160"— 50-223 5"+ 0-596"= 17 325 593-988 5"

and adding the secular acceleration J

n"= 17 325 593-985 8''+o-2i64''(^^) (26)

It is easily seen that

and also

n" = ^(g + co-&-^) (27)

nf^z^g + cD-v'+x) (28)

But when the numerical values of the quantities in the right hand members of these
equations are substituted from Hansen's Tables de la Lune, pp. 15 and 16, a discord-
ance appears in the terms involving the square of the time; equation (27) giving for
the coefficient +0-00033260", while equation (28) gives +000040419". Taking
the mean of these two numbers, Hansen's value of the mean sidereal motion of the
Moon in 365^ days is

// // , r "ft — i8oo\

n — 17325 593'973 * +0-24360 ^- J

. t:o 11ft— i8oo\
+ 000036840 ( J

«-i8ooV (29)

In his "Researches on the Motion of the Moon," Newcomb found that Hansen's
tables required the correction §

- ri4" -0-291 7" (*- i8oo)-3-86"(^~l8ooY

*5o. p. 3- t44, P-io. t44, P- 8. § 6o, pp. 268 and 274.

H

ON THE SOLAR RARALLAX AND ITS RELATED CONSTANTS.

to the mean longitude of the Moon; and the correction + o*io" (t — 1800) to the motion
of the Moon's node. Or, in other words, if we put g0, g'0, go0, go'0 and &0 for Newcomb's
corrected values of the quantities g, g', go, go', and ©, then

^=//-i-i4'/-o-29i7'/(^-i8oo)-3-86''(^I^)2

GOQ — GO — O'lO" (t— ISOO) (30)

co'0 — go' — o* i o" (t — 1 800)

0O -O —OIO" (£— I800)

and the substitution of these corrected values in equations (27) and (28) gives

n" =*t(g + Co-0-zpl)-o-29i7" -0-0772 (^^) (3i)

d , . / , n // st— 1 8oo\ f N

= jt(g + oo-oo'-\- X) -0-2917 -0-0772 ( ioq j (32)

Newcomb's value of the mean sidereal motion of the Moon in 365^ days is there-
fore

(33)

n" = 17 325 593-68: 4" + 0166 40" (j-^)

/:o // ft— l800\2
+ OOOO3684O ( J

If we put

n" = a + bt + ct2 (34)

the expression for Tlf the length of the sidereal month, will be

T.=36tf£.3y = 473364oo°" \ ,-*<+(*;-«>-«*■ 1 05)

a+o£ + cr a ( a Vflr ay >

to terms of the third order in (£ — 1800).

Formulae (26), (29), and {t>Z) are 0I> tne same form as (34), and the several values
of T: are to be determined from them bv substituting in (35) the values of a and b
which they contain. In view of the uncertainty as to the true value of the secular
acceleration of the Moon's motion, the terms depending upon the square of the time
can have no real significance, and will be neglected.

From (26) and (35), according to Airy

Tjzz 27-321 66o682d — coooooo 341 25d( — j

= 2 7do7h 43m 1 1 -483s - 0-029 484s (l ~ 1 8o° )

V 100 J

ON THE SOLAR PARALLAX AND IIS RELAT1 D < ONSTANTS. 15

From (29) and (35), according to Hansen

T1=. 27321 660 702'' — 0-000000384 i5df -J

^ 1800 \ (37)

-27do7h 43m n-4848-oo33 191*^ ^ J

And from (33) and (35), according to Newcomb

Tx = 27-321 661 i62d — o-oooooo 262 4od( — j

z= 2 7d 0711 43m 1 1 -524s — o-02 2 67i8( — \

Equations (36), (2,7), and (38) yield the following values for the mean length of
the sidereal month at the epoch 1850-0, expressed in mean solar seconds

Airy 2 360 591-468"

Hansen 2 360 591-467 (39)

Newcomb 2 360 591-513

For the mean svnodical, tropical, nodical, and anomalistic motions of the Moon
we have

ni" = ^-g' + C0-00') -0-2917" -00772" 0=^) (4o)

<=|(j+fi,-e)-o-29i7"-oo772"(^^2) (40

*3" = I (9 + •) -0-3917" -00772" (tj=) (4»)

„ « = I W - o-29 . 7" - 00772" (^5) (43)

where in each case the literal part of the expression is Hansen's value, and. the numeri-
cal terms are Newcomb's corrections. Of these expressions the most important is (40).
Hansen's value for it is

nl' = 16 029 616-5331" + 0-243 824" ( ^q00)

,, 1 t — i8oo\

m, (44)

11 si — i8oo\

n /l — iouu\

+ 0-00040419 (- IOO-J

and Newcomb's value is

// rr /- II ft iSOON

11,"= l6 029 6l6-24I4 +OI66 624 ( )

(45)

+ 0-00040419 (^ ioq j

i6

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

(46)

(47)

Substituting- these values of w/' in formula (35), and neglecting the terms involv-
ing the square of the time, the mean length of the synodical month comes out,
according to Hansen

29-530 587 898d — o-ooo 000 449 1 9d ( — -^

= 29d i2h 44m 02794a-o-038 8io8^~l8o°^)

and according to Newcomb

29530 588 435d — o-ooo 000 306 96d f — \

— 29d i2h 44m 02-841" — 0*026 522s ( — j

As a check on the preceding computations, we may employ the relation

Tx (T + T2) = TT2 (48)

where T, Tx and T2 are respectively the lengths of the sidereal year, the sidereal
month, and the synodical month. With the values of these quantities from (16), (38)
and (47), we find

T^T + TOzz 10 786-235 1 74 5 days
TT2 zz 10 786-235 1 "J2> 6 days

For the other months, at the epoch 1850-0, we have from equations (41), (42),
and (43), by substituting the values of g, go, and © from Hansen's lunar tables, and
neglecting Newcomb's corrections

%"= 17325644-33'
*h"= 17 395 27364

iu" — 17 179 158-87

(49)

whence

Mean Tropical Month =27-321 581 292* zz 2fx ojh 43™ 04-624"

Mean Nodical Month — 27-212 219 238d zz 27d 0511 05™ 35742s (50)

Mean Anomalistic Month zz 27554 550463d zz 2 7d 1311 i8m 33- 1608

9.— THE CONSTANTS M, ™, (1 + «'), e2, y, AKD I, PEETAIXING TO THE MOON.

For the regression of the Moon's node relatively to the line of equinoxes, in 365^
days, Hansen's lunar tables give *

d

.,, st— 1800 x

^ = ~^(0 + ^) = -69679'6l9I''+O"I4I36,'(^Too~) (5°

Whence, for the epoch 1850-0

/^zzi9° 21' 19-5484" zz 0337 815 984 of radius (52)

*54, PP- 15-16-

ON THE SOLAR PARALLAX \NI> ITS RELATED CONSTAN I 7

Or, with Professor Newcomb's correction, from (30)

ju=iq° 21' 194484" = 0337 815 499 of radius (53)

The ratio of the mean sidereal motions of the Sun and Moon results immediately
from (39) and (16), thus

m=S= 2360 59i-5i3; = 0.07 8 II2 (54)

By putting

0^ = ^(1 + x')

and comparing this with the expression given by Delaunay for the constant part of
the lunar parallax, namely*

i=ij,+('+u«y-is*'-«rf| (5s)

rx (U ( \o 4 ; 255 40 >

we find

■+"'='+(>;'■■ )'"=-^"',-48'"s (56)

Whence, by substituting the values of el and m from (24) and (54)

1 + *' = 1 'ooo 908 743 (57)

From all the lunar observations made at Greenwich between the years 1750 and
1847, inclusive, Sir G. B. Airy found for the coefficient of the first term of the equa-
tion of the Moon's center f

22 639-06" (58)

Professor Newcomb found that for the same term Hansen used in his lunar
tables %

2264015" (59)

and from all the meridian observations of the Moon made at Greenwich between the
years 1847 and 1874, and at Washington between the years 1862 and 1874, inclusive,
Newcomb found the correction —0-57" to (59), § which thus becomes

22639-58" (60)

Taking the mean of (58) and (60), giving the former double weight, and equat-
ing the result to its analytical equivalent,|| we obtain

22 639-233' = 2 r, - 1 e2a + ^ ef-

*52, T. 2, p. 914. Compare also 63, T. I, pp. 664 and 674. ?6i,p. 29, and 62, p. 69.

1 44, p. 13. 1 62, P- 69. II 52. T. 2, p. 804, equation (7).

6987 2

!^ ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Whence

e2— 1 1 323880" =0054 899 720 of radius (61)

For the coefficient of the first term in the development of the Moon's latitude,
expressed as a function of the Moon's true longitude, Airy found from the Greenwich
observations made between the years 1 750 and 1847, inclusive,* 18 535-55"; but when
expressed as a function of the time, this becomes, according to Delaunayj

18461-26" (62)

Professor Newcomb found that the value of the same coefficient implicitly con-
tained in Hansen's lunar tables is J 18 461 629" ; and to that Newcomb found a correc-
tion of —015" from the Greenwich and Washington observations of the Moon made
between the years 1862 and 1874. § His corrected value is therefore

18461-48" (63)

Taking the mean of (62) and (63), giving the former double weight, and equat-
ing the result to Delaunay's analytical expression for it,|| we have

1 8 46 1 -33" r= 0-089 503 054 of radius

— 2/ — 2/tv — - ys + 7 ye2* + - y*e£ 5 - ye*

4 32 4 H4

whence, with the value of e2 from (61)

y — 0044 886 793 (64)

As y is the sine of half the inclination of the Moon's orbit to the plane of the
ecliptic, (64) gives

1 = 5° 08' 43-3546" =0089803 757 of radius. (65)

10.— OBSERVED VALUE OF THE PARALLACTIC INEQUALITY OF THE MOON.

From the nature of the case, the observations for determining the coefficient of
this inequality must be made partly upon the first, and partly upon the second limb
of the Moon, and thus they involve all the systematic errors which may arise from the
different conditions under which these limbs are observed, and all the uncertainty
which attaches to our knowledge of the Moon's semi-diameter. The following values
are perhaps the best hitherto obtained :

1. From his discussion of the lunar observations made at Greenwich between the
years 181 1 and 1851, Sir G. B. Airy concluded that the most probable value of this
coefficient is 1 24-7" H

2. Professor Newcomb found that the value deduced by Hansen from the Green
wich and Dorpat observations is 126-46" **

* 44, p. 21, and 45, p. 27. J 62, p. 76. ||52, T. 2, p. 862, equation (1 ). ** 232, p. 23.

1 52, T. 2, p, 802. g6i,p. 36. tf44,p. 16.

ON" THE SOLAR PARALLAX AM) ITS RELATED <<>NST.\NIS

19

3. From 2075 lunar observations made at Greenwich between the years 1848
and 1866, Mr. E. J. Stone found 1 25 36" ^ 04" ; the probable error being estimated.*

4. From the lunar observations made at Washington between the years 1862 and
1866, Professor Newcomb found 125-46". f

5. From an extended discussion of the whole subject, Messrs. Campbell and
Neison found, cither 1 25 64" ±009", or 12464" ±0*2 5", according as a certain hypo-
thetical 45-year term was or was not admitted into the lunar theory.J

6. From a comparison of Hansen's lunar cables with some 1 600 observations of
the Moon, made with the Greenwich transit circle between the years 1862 and 1877,
Mr. Edmund Neison found 1 253 1 3" ±0-046" ; but he thought it probable that that
value might require diminution by 073" on account of the before-mentioned hypo-
thetical 45-year term.§

The controversy carried on in the Monthly Notices of the Royal Astronomical
Society by Mr. Stone and Messrs. Campbell and Neison, during- the years i88o-'82,
shows that the entire mass of existing lunar observations must be thoroughly redis-
cussed before a definitive value of the parallactic coefficient can be obtained. Respect-
ing- the data given above, it may be remarked that the values in paragraph 5 are
superseded by that in paragraph 6, and that the mean of the values in paragraphs 1
and 2 is nearly the same as the values in paragraphs 3, 4, and 6. All questions of
weights may therefore be disregarded, and by taking the arithmetical mean of all the
values except those in paragraph 5, we find

Q= 125-46" ±0-35" (66)

where the probable error is estimated to be one-fifth of the difference between the
greatest and least values.

11.— OBSERVED VALUE OF THE LUNAR INEQUALITY OF THE EARTH.

The magnitude of the coefficient of this inequality is only about two-thirds that
of the solar parallax, but as it depends upon differences of the sun's right ascension,
which is always observed in precisely the same way, it should be free from constant
errors, and can therefore be determined with great accuracy. The following are the
best available data :

From observations at Greenwich, Paris, and Koenigsberg. made during the periods
stated, Le Vekeiek found ||

Greenwich 18 16-1 826 L =: 6-45"

Greenwich 1 827-1 850 656

Paris 1 804-1 8 14 661

Paris 18 1 5-1 845 647

Koenigsberg 18 14-1830 643

Mean = 6-50 ± 0023'

'65, p. 271. t232, p. 24. t46, p. 467- §59. P- 409- ||4i,l>ioo.

20 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Professor Newcomb found the following additional values*

Greenwich 1 85 1-1 864 L = 6 56" ± 004"

Washington 1861-1865 6-51 ±007

Giving- these three results weights inversely proportional to the squares of their
probable errors, namely. 9 26, 3 06, and 1, we obtain

L- 65 14" ±0016" (67)

lli. — OBSERVED VALUE OF THE LUX Alt PARALLAX.

For the determination of this constant, all the data at present available are based
upon declinations of the Moon, observed respectively in Europe and at the Cape of
Good Hope.

From a comparison of Lacaille's observations at the Cape with those made at
Greenwich, Paris, Berlin, and Bologna, during the same period, namely, from June,
1 75 1, to February, 1753, Professor Olufsen found f

x = sin P — 0016 512 t,^ +0024 492 01 £ — 0000 001 62 dh

or, multiplying by 3423 3"/ sin 3423 3" = 206 274-28

P = 3406069" + 5052-072" e — 0334" dh ± 0.45" (68)

where £ is the Earth's compression, and dh the error in the longitude of the Cape,
expressed in minutes of time. Neglecting dlj, we have from (4)

293466
whence

P = 3406 069" +17-21 5' = 3423284' ± o 45" (69)

From his own observations of the Moon at the Cape, combined with observations
made at Greenwich and Cambridge during the same period, namely, from May, 1832,
to May, 1833, Professor Thomas Henderson deduced two values of the lunar parallax ;
one by comparison with Burckhardt's tables, and the other by comparison with
Damoiseau's tables. As Burckhardt's parallaxes are now known to be erroneous, we
have only to consider the result from Damoiseau's tables, which was J

where

P rr 342246'' + 5062" 5c — 0-05" St — o'i2 6s — 014 S.s' (70)

Longitude of Cape Observatory ~ 1 h i$m 55s 4- r5^8

Sc = £ —

300

1 232, pp. 25-26. 1 74. P- 226. j 73, p. 294.

ON ["HE SOLAR PARALLAX AND ITS RELATED CONSTANTS. 21

and Ss and Ss' are corrections for any constant differences which may have existed in
the values of the Moon's semi-diameter given by the different instruments, and the
different observers.

Neglecting Ss and Ss'} we have

St = -o-3sa

* i i

OC Z= — — = 0-000 0/4 2 2

293'466 3OO

Whence, from (70)

P = 3422-46" +0-376" +0-018" = 3422-854" (71)

By combining 123 observations of the Moon made at the Cape during the years
1830 to 1837, with corresponding observations made at Greenwich, Edinburgh, and
Cambridge, Mr. Breen found*

P =23422-696" — 0*013 St (72)

where St is the correction to the assumed longitnde of the Cape, and the compression
of the Earth is taken to be 1/300. With our value of the compression, namely,
1 / 293-466, and St = — o-35s, (72) gives

P = 3422-696" 4- 0-376" + 0-005" = 3423-077" (73)

Bv combining 239 observations of the Moon made at the Cape during the years

1856 to 1 86 1 with corresponding observations made at Greenwich, Mr. E. J. Stone

found f

P — 3422-707" ± 0-049"

Mr. Stone does not state what compression he employed, but as 1/300 was then
used both at Greenwich and the Cape, the same correction should probably be applied
as in (71) and (73). That gives

P = 3422-707" 4- 0-376" = 3423-083" ± 0-049" (74)

Collecting our results, we have from (69), (71), (73) and (74),

P=: 3423-284' according to Olufsen.

3422-854 according to Henderson.

3423-077 according to Breen

3423*083 according to Stone.

When it is remembered that the first of these values is based upon observations

made with old-fashioned quadrants, the second and third upon observations made with

mural circles, and the fourth upon observations made with large transit circles, their

agreement is remarkable. However, as the observations upon which the second result

rests have all been embodied in the third, we will adopt the mean of the last two. and

put

P = 3423-08" ±0050" (75)

If we assume the probable errors of the latitudes of Greenwich and the Cape to
be ±0-05". then the probable error of P should be increased to ±o-i 21"

*72,p. 137. t75. p- 16.

22

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

13.— THE CONSTANT OF PRECESSION.

Putting ip and ipx respectively for the secular parts of the luni-solar, and of the
general precession; go for the mean inclination of the equator of 1850 -f- 1 to the fixed
ecliptic of 1850; and col for the mean inclination of the equator of 1850 + / to the
ecliptic of 1850 + ^; then, according to Le Verrier and Serret, we shall have*

( 50371 Ao"+x + ooiAS"y + 0-1743" "' h_o-oooio8 8o6"^

( + 00169" v"1 —0-057 5" v1T — o-oi2 4 ' vv )

go — &>„ + 0-000007 180" t2 (76)

ip! = (50-235 72" + x) t + o-ooo 1 1 2 900 t2

( 0-475 66" +0-005 3" k + 0-288 8"^ I *

a>. — &>0— < ^/J „ ' ... ^ ' . „ }t — o-ooo 001 490 t~

X +0-008 3" ^" + 01601"^+ 0-013 1" vx s

where a; is a small unknown correction to the precession constant, and t is counted in
Julian years from the epoch 1850-0. From the adopted masses of the planets, given
on pages 42 and 48, we have

v =-0655353
y' rr — 0^007 004

vm = — 0-133558

vlT:= + o-oo2 339

vv — + o-oo2 970

Whence, by substitution in (76)

ip = (50-358 25" + ./) t — o-ooo 108 806" e

go =: oo0 + OOOO 007 1 80" t2

Xp-L = (50-235 72" + X) t + OOOO I I 2 9OO" t2

a>! = go0 — 0-46947" f-o'oooooi 490" t2

(77)

From the series of observed values of the obliquity of the ecliptic given by Le
Verrier,! and with the theoretical value just found for the annual change of the
obliquity, we deduce for 1850-0

G>0=23° 27' 31-36" ±0-345" (78)

To find the planetary precession, A, we have

^ — ^1 = A COS £ (GO + GJj) (79)

where it will be sufficiently accurate to take

cos %(go + g>i) — cos (a>0 — 0234 735" t) = cos &>„ + 0-234 735" tare 1" sin co0 (80)

The substitution in (79) of the values of if>, rf>u go and cou from (77), (78) and
(80), gives

Whence

0-122 53 / — oooo 221 706" t2— A (0917 347 2+0-000000453 t)

A— 0133570" t — 0000241 748" t2

(81)

*8, T. 2, 1 . 174, and 83, p. 324.

t4i. p. 51.

ON THE SOLAR PARALLAX AND IIS RELATED CONSTANTS.

Equations (77), (78) and (81) give

' ^ =50-358 25" +.r — 0-000217612" t
' ?? = 5°'235 72" +.r + o-ooo 225 800" t

' (U — o-I33 57" -0-000483 496" t

sin co = 0398 088 1 2 -f 0000 000 000 03 1 9 t2
cos co = 0-9 f 7 347 1 7 — o*ooo 000 000 01 3 9 £2

And by substituting these values in the well-known expressions

dd> dA (lib .

m = — cos «>— - n=z ,- sin ©

dt dt dt

23

(82;

we obtain

m — 46-062 43" -f o-ooo 283 870" t -f 091 7 347 #
w z= 20-047 °2" — o'ooo 086 629" t + 0-398 088 a?

(83)

For the epoch 17775, from a comparison of his own reduction of Bradley's
observations with Piazzi's catalogue for 1800, Bessel found*

m zz 46-034 002" n = 20*064 472

When Newcomb's correction for systematic errors in the right ascensions of the
catalogues is applied,! namely, — o*43"/45 zz — O'ooo 555", these numbers become

«r 46*024447" fizz 20-064 472

and their substitution in (83), together with t — — 72*5 years, gives

from m, xzz — o*oi 8 97" from n, xzz-\- 0*028 06

Giving the result from m double weight, and taking the mean

#r= — 0003 29" (84)

For the epoch 1 790, from a comparison of Bessel's reduction of Bradley's obser-
vations with the Dorpat observations, Otto Struve found the general precession to be
50*234 92" i 0*007 57". % Applying Newcomb's correction for systematic errors, §
namely, —0*3 7" / 70 zz — 0*005 29"> tne precession becomes 50*22963". Formula
(82) gives for the same epoch 50*222 17", and therefore

£ = + 0*00746" (85)

For the epoch 18447, fr°m a comparison of AVeisse's reduction of Bessel's zones
with Schjellerup's catalogue, Nyren found || w = 46*026 5". Applying Newcomb's

♦76, p. 404. t^i.p. 108. +"85, p. 104. §81, p. 108. ||82, p. 571.

24

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

correction for systematic errors, namely,* + ro4"/36 -= +0*028 9", this; becomes
m = 46*055 4", which being substituted in (83), together with £ = — 5-3 years, gives

r — — o 006 03'

(86)

For the epoch 1829-7, from a comparison of Lalande's zones' with Schjellerup's
catalogue, Dreyer foundf m — 46-066 6", which being substituted in (83), together
with t = — 20-3 years, gives

x = + 0-01082" (87)

For the epoch 18050 Mr. Ludwig Struve made a comparison between Dr.
Auwer's reduction of Bradley's observations and the Pulkowa catalogue of 1855,
from which he found J

m rr 46*041 7" + o-ooo 274 1" t

where t is reckoned from the year 1800. His observed result for 1805-0 must there-
fore have been

m z=z 46 043 07"

and the substitution of that in (83), together with £ = — 45*0 years, gives

x = — 0*007 1 8"

(88)

Mr. L. Struve speaks highly of Dr. Bolte's Untersuchungen fiber die Priicessions-
constante,§ but the present writer has never seen that work.

Collecting our results, we have from numbers (84), (85), (86), (87), and (88)

Authority.

Value of x.

General Precession.

1800.

1850.

Bessel

O. Struve

Nyren

Dreyer

L. Struve

Means

//
— 0-003 29
+0-007 4°
— 0-006 03
+0010 82
— 0-007 J8

//
50-221 14
50231 89
502 1 8 40
50-235 25
50-217 25

50-232 43
50243 18
50- 229 69
50- 246 54
50-228 54

-fo-ooo 36 50-224 79

50-236 08

And by substituting the mean value of x in (yy)

ip =(50-358 61" ±0-00248") ^- o-ooo 108806" t2

^ — (50-236 08" ± 0-002 48") t + O-OOO I I 2 900" t2

(89)

*8i, p. 109.

177. !>• 154-

+ 86, ]>. 30.

jJ An inaugural dissertation, Bonn, 1883.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. 25

14.— THE CONSTANT OF NUTATION.

The following- is an abstract of the most important determinations of the constant
of nutation hitherto made :

// //

1821. BRINKLEY, from 1618 zenith distances of 10 stars, measured at the observatory of Trinity College, Dublin,

between the years 1S08 and 1820. (90, p. 347.) 9-25 ^0-05

1836. Busch, from 1949 zenith distances of 23 stars, observed at Kew and Wansted by BRADLEY during the

years 1727 to 1747. (91, p. ^S.) 9-232^0031

1838. Robinson, from 6023 zenith distances of 15 stars, measured at Greenwich with the mural circle during

the years 1812 to 1835. (113, p. 18.) 9-239 -J- 0052

1841. Lundahl, from more than 1200 zenith distances of Polaris, observed at Dorpat during the years 1822 to

1838. (99K'P- 33) 9-236 J- 0-040

1 84 1. C. A. F. Pf.ters, from 603 right ascensions of Polaris, observed at Dorpat by Struve and Preuss during

the years 1S22 to 1838. (109, p. 161.) 9-216 -J-0020

1855. Main, from 173 zenith distances of y Draconis, observed at Greenwich with the 25-foot zenith tube during

the years 183710 1847. (102, p. 1S6.) 9'323i°'°S9

1868. E. J. Stone, from 3250 zenith distances of Polaris, 51 Cephei, and 6 Ursx- Minoris, together with 1936

right ascensions of Polaris, all observed at Greenwich, with the transit circle, during the years 1 85 1 to

1867. (114, p. 249.) 9-134 -i- 001 1

187 1. Nyren, from 375 observations of ti Ursae Majoris, 1 Draconis, and oa Draconis, made at Pulkowa with the

prime vertical transit instrument, during the years 1840 to 1862. (104, p. 30.) 9-244 -J- 0-0I2

1882. Downing, from 1041 zenith distances of y Draconis, observed at Greenwich with the reflex zenith tube,

during the years 1857 to 1875. (92, p. 344-) 9'335 ±0-032

1885. De Ball, from 1867 right ascensions of Polaris, 51 Cephei, and A Ursae Minoris, observed at Pulkowa by

Wagner during the years 1861 to 1872. (89, p. 42.) 9-217-^-0012

The probable errors attached to Brinkley's and Robinson's results have been
taken from Peters's paper,* and that attached to Stone's result has been computed by
the present writer.

Giving- the determinations by Main and Downing half weight, because they rest
upon a sing-le star, we have from the weighted mean of the whole series

$ = 9-2331" ±0-0112" (90)

Busch's determination should probably have been rejected on account of the
errors discovered in his computations by Dr. AuwERS,f but its retention does not sen-
sibly affect (90). The probable error of (90) is largely increased by the constant
errors which evidently exist in the results found by Main, Stone, and Downing ; but
as their determinations rest upon more than 6400 observations, made with three differ-
ent instruments, it does not seem prudent to ignore them.

15.— THE CONSTANT OF ABEERATION.

The following is an abstract of the best values hitherto obtained for the constant

of aberration :

// /'

1817. Bessel, from his discussion of Bradley's observations, (i, p. 123.) 20-475

1819. Piazzi, by the observations made at Palermo (5, p. 207) 20-229

1821. Br inkley, from 2633 zenith distances of 14 stars observed at Trinity College, Dublin. (90, p. 350.) . 20-372
1S22. F. G. W. Strive, from 693 differences of right ascension between 6 pairs of stars observed at Dorpat.

(115, p. lxiv.) 20349

This result was subsequently corrected by C. A. F. Peters to 20-361" -J- 0-OIS6". See ill, p. 55.

•109, p. 132. |88, p. 611.

26 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

// //

1828. RICHARDSON, from 41 19 zenith distances of 14 stars, observed at Greenwich. (112, p. 68.) 20-503

1836. BUSCH, from 1949 zenith distances of 23 stars, observed by Bradley at Kew and Wansted. (91, p. 338.) 20-212 -J- 0-038

1839. Henderson, from 231 zenith distances of Sirius, observed at the Cape of Good Hope. (95, p. 248.) . 2041

184 1. F. G. W. STRUVK, from 19 observations of v Ursa- Majoris in the prime vertical, at Pulkowa. (1 16, p. 290.) 20-493 ±0-040
1841. Lindenau, from 800 right ascensions of Polaris, observed at Greenwich, Konigsberg, Dorpat, Palermo,

Milan, and Seeberg, during the years 1750 to 1816. (98, p. 62, and 1 1 1, p. 65.) 20-449 -J- 0-032

1841. C. A. F. PETERS, from 603 right ascensions of Polaris, observed at Dorpat. (109, pp. 142 and 180.) . . 20-425 -J- 0-017

1841. LiNDAiii, from more than 1200 declinations of Polaris, observed at Dorpat. (99}4>P-37-) 20-551-1-0.043

1842. Henderson, from 272 double altitudes of a1 and os Centauri, observed at the Cape of Good Hope. (96,

P-370-) 20-523 -J- 0-065

1843. F. G. W. Struve, from 298 observations made upon 7 stars with the prime vertical transit instrument at

Pulkowa. (118, p. 275.) 20445 ±0-011

1849. C. A. F. Peters, from Bradley's sector observations at Greenwich, (in, p. 23.) 20-522 -[-0-079

1849. C. A. F. Peters, from 704 declinations of 8 stars observed with the Ertel vertical circle at Pulkowa.

(in, p. 138.) 20-481^0013

1849. Lindhagen, from 396 right ascensions of Polaris, observed at Pulkowa. (99, p. 354.) 20498 -|- 0-012

185 1. MACLEAR.from 391 double altitudes of n1 and a* Centauri, observed at the Cape of Good Hope. (100,

P- 98) 20531 ±0-038

1852. Maclear, from 137 double altitudes of (i Centauri, observed at the Cape of Good Hope. (101, p. 152.) 20-594 -(- 0-049
i860. Main, from 486 zenith distances of y Draconis, observed at Greenwich with the reflex zenith tube, during

the years 1852-1859. (103, p. 190.) 20-335 -[- 0-023

1882. Downing, from 1041 zenith distances of y Draconis, observed at Greenwich with the reflex zenith tube,

during the years 1857-1875. (92^.344.) 20-378^0-040

1883. Nyren, from the series of observations made at Pulkowa, with the vertical circle, by Peters, Gylden,

and Nyren ; with the transit instrument, by Schweizer and Wagner ; and with the prime vertical

transit instrument, by F. G. W. Struve and Nyren. (106, p. 47.) 20-492 _[- 0-006

1888. A. Hall, from 436 observations of a Lyroe, made with the prime vertical transit instrument at Washing-
ton, during the years 1862-1867. (94, p. 12.) 20-454 -J- 0-014

1888. KOstner, from 244 differences of meridian zenith distance of 7 pairs of stars, measured with the uni-
versal transit at Berlin. (97, p. 45.) 20-313 -j-o-oi 1

A clearer exhibition of the facts will be attained b}r arranging the foregoing data
somewhat differently, and in so doing Peters's reductions of Bradley's observations
at Greenwich and Struve's observations at Dorpat will be accepted to the exclusion
of those by Bessel and Struve himself; the determinations by Piazzi and Brink-
ley will be omitted, the latter on account of the unexplained systematic errors exhib-
ited by his declinations ; Busch's reduction of Bradley's observations at Kew and
Wansted will be omitted on account of the errors discovered in Busch's computations
by Auwers;* Struve's preliminary result from his observations of v Ursse Majoris in
the prime vertical will also be omitted because the same observations are embodied in
his final result, published in 1843; and Nyren's result will be separated into its original
constituents. When these changes are effected, and the results are classed under the
observatories furnishing the observations, we have the following exhibit:

GREENWICH.

//

Peters, from Bradley's sector observations 20- 5 22

RICHARDSON, from mural circle observations 20503

Main and Downing, from zenith distances of y Draconis ■ . . . . 20-356

CAPE OF GOOD HOPE.

Henderson, from Sirius 20-41

Henderson and Maclear, from a Centauri 20-527

Maclear, from (3 Centauri 20594

*88, p. 611.

ON THE sol.AR PARALLAX AND [TS RELATED CONSTANTS. 27

DORPAT.

//

F. G. W. STRTJVE, from transit observations 20-361

LuNDAHL, from declinations of Polaris 20550

C. A. F. Peters, from right ascensions of Polaris 20425

PULKOWA.

F. G. W. Strtye, from prime vertical" observations 20-445

C. A. F. Peters, from vertical circle observations 20481

LlNDHAGEN, from transit observations 20498

Nyren, from Gylden's observations with the vertical circle ... 20-469

Nyren, from his own observations with the vertical circle .... 20-495

Nyren, from Warner's observations with the transit 20-483

Nyren, from his own observations with the prime vertical transit 20517

BERLIN.
Kustn'ER, from the zenith-telescope method 20-313

WASHINGTON.

A. Hall, from observations of a Lyne with the prime vertical transit 20-454

VARIOUS OBSERVATORIES.
Lindenau, from right ascensions of Polaris 2C449

Unmistakable evidences of constant errors are exhibited in these values of the
constant of aberration — most notably in those found by Main, Downing, Maclear
from ft Centauri, F. G. W. Struve at Dorpat, and Klstner. Nevertheless, it is diffi-
cult to assign thoroughly satisfactory reasons for rejecting any of them. The Pul-
kowa values are probably the most correct, but even they exhibit a range of 0-072".
Perhaps no two astronomers would assign the same relative weights to the different
determinations, and yet, within rather wide limits, it is precisely these weights which
determine the magnitude of the final result If we take the means by observatories,
we find

Greenwich 20-460) Berlin 20313

Cape of Good Hope .... 20-510 Washington 20-454

Dorpat 20-445 Miscellaneous 20449

Pulkowa 20-484

and the arithmetical mean of the results from all the observatories, except Berlin, is

a zz 20-467" ± 0-007"

Again, if we take the arithmetical mean of all the results in the last general
exhibit, tin; Pulkowa values will have relatively a little more weight than the others,
and we shall obtain

a — 20-466" ±o-oii" (91)

which will be adopted. This lies almost exactly midway between Struve's classic
value and that recently found by Nyrlx in his admirable paper on the aberration of
the fixed stars.

2g ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

16.— TEE LIGHT EQUATION.

The time occupied by light in traversing the mean radius of the Earth's orbit is
usually called the light equation, and there are but two determinations of it from the
eclipses of Jupiter's satellites, namely, Delambre's, published in 1792, and Glase-
napp's, published in 1874.

Delambre's value is 4932s. which he originally derived from 500 eclipses of
Jupiter's first satellite, and subsequently revised without obtaining any sensible cor
rection, although he used more than a thousand eclipses of the same satellite. The
details of these investigations have never been published, and our knowledge of them
is confined to the brief allusions contained in Lalande's Astronomy,* Delambre's
Astronomy,f and Delambre's Tables e'cliptiques des satellites de Jupiter.} The last is
the most explicit.

Glasenapp's value is 50084s i ro28,§ which he derived from 391 eclipses of the
first satellite of Jupiter, observed during the years 1 848 to 1 8 jt,. His memoir, although
very valuable, is rendered almost inaccessible by being printed in the Russian lan-
guage; but Mr. Downing has done something to remove that obstacle by publishing
an excellent account of the work in The Observatory, vol. 12, pp. 173 and 210.

In combining these two results the following facts must not be overlooked :

1. The eclipses used by Delambre were inferior to those used by Glasenapp on
account of having been observed with less powerful telescopes, and possibly with less
accurate knowledge of local time; but their inferiority can not have been great, because
Delambre says of the eclipses he used, "il n'est pas rare de voir deux observations d'une
mime eclipse diffe'rer entre elles dime demi minute, "|| and residuals of that magnitude
are not rare among Glasenapp's equations.

2. After a thorough trial of Bailly's photometric method of correcting observa-
tions of the eclipses of Jupiter's satellites, both Maskelyxe and Delambre abandoned
it as useless;!! and yet Glasenapp's investigation is founded upon that very method,
with some modifications whose value it is difficult to estimate. However, it will not
escape notice that with the application of all his corrections Glasenapp found the light
equation to be 500,84srb L028, while without them he found it to be 497 15s =b 1 208.
The diminution effected in the probable error by the appli cation of the corrections is
so small as to indicate either the failure of Bailly's method or the existence of periodic
errors in the tables of the motions of the satellites. Glasenapp thought the latter
hypothesis the more probable.

3. Delambre's result depends upon more than a thousand eclipses, while Glase-
napp's depends upon onlv 391. And here we encounter the singular circumstance that
the result which Delambre obtained from 500 eclipses was not sensibly modified when
he used more than a thousand; while Glasenapp's result, which rests upon nearly 400
eclipses, differs largely from Delambre's. The existence of constant errors in one or
both series of observations seems the most probable explanation.

In view of these facts, it is not clear that Glasenapp's result is entitled to more
confidence than Delambre's; nevertheless, we adopt the arithmetical mean of the two,

*6,T. l,Tal)lesastronomiques,p. 238. J 127,]). vij. J,T. 3,p. 502.

t.i.T-.i-PP- lo5- 1 06 and 502-507. $129, p. 131, and I28,p.2ll. ^ 3, T. 3, p. 507; 4, p. 746, and 12, v ol. 1, p. 266, sec. 464.

ON I HE SOLAR PARALLAX ^ND ITS RELATED CONST A]

2<>

thus giving Glasenapp rather more than double weight, and in view of tin; uncertainty
of the result we attribute the whole of Glasia ait's probable error to it. In thai way
we find

Light equation, 9 = 497*0" rb L028 (92)

which is almost identical with the result obtained by Glasenapp when he omitted his
corrections to the observed times of the eclipses.

17.— V, THE VELOCITY OF LIGHT IN VACUO.

The velocity of light can be measured between points upon the Earth's surface,
either by the toothed-wheel method or by the revolving-mirror method. Both methods
have been used, and the following are the principal results:

Fizeau found

V = 70 948 lieus of 25 to a degree m 3 1 5 324 kilometers =. 1 95 935 miles

His experiments were made in 1849 by the toothed- wheel method, working across
an interval of 8 633 kilometers = 5*364 miles, between Suresnes and Montmartre,
Paris.*

Foucault's experiments gavef

V z= 298 574 =1= 204 kilometers— 185 527 i 127 miles

He used the revolving-mirror method, and worked across an interval of only 20
meters = 65*6 feet, at Paris. His experiments were in progress from May to Septem-
ber, 1862; but he based his final result upon the 80 observations made on September
18, 19, and 21.

Cornu found, from the experiments which he made in August, 1872,

V = 298 500 zb 995 kilometers = 185 481 d= 618 miles

This result rests upon 658 experiments, made by the toothed-wheel method, work-
ing across an interval of 10310 kilometers = 64064 miles, between l'Ecole Polytech-
nique and Mont-Valerien, Paris. J

Cornu found, from the experiments which he made in September, 1874,

V — 300 400 rh 300 kilometers = 1 86 662 i 1 86 miles

This result rests upon 546 experiments, made by the toothed-wheel method, work-
ing across an interval of 22910 kilometers = 142357 miles, between the observatory
and Montlhery, Paris. §

Michelson found, from his experiments of June and July, 1879,

V zz 299 910=1= 51 kilometers = 186 357 ±317 miles

This result rests upon 100 experiments, made by the revolving-mirror method,
working across an interval of 1986*23 feet = 0*3762 of a mile = 0*6054 of a kilometer,
at the U. S. Naval Academy, Annapolis, Md.||

*I32,P.92. ti3S.P-224. } 130, p. 178. gi3i,p.A.298. ||I37. P- I57,andi39, p- 244.

30 "M I UK SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Young and Forbes found

V = 301 3 84 ± 263 kilometers zz 187 273 ± 164 miles

This result rests upon only 12 experiments, made in December, 1880, and Jan-
uary, 1 88 1, by a peculiar application of the toothed-wheel method. The light from
the collimator containing the toothed wheel was sent simultaneously to two reflecting
collimators situated at different distances, but nearly in- the same straight line; and the
observation consisted in determining when the images returned by these two collima-
tors were of equal brightness. The toothed-wheel collimator was situated at Kelly
House, Wemyss Bay, Scotland, and the reflecting collimators were located on the hills
behind Innellan, across the mouth of the river Clyde, their distances being respectively
3-1884 and 3 4493 miles, or 5-13 13 and 5*5510 kilometers.*

Newcomb found three results from the experiments which he made at Washington
by the revolving-mirror method during the years 1880 to 1882. They are as follows:!

(«) From 148 experiments, made between June, 1880, and April, 1881, across an
interval of 5*1019 kilometers — y 1702 miles, between Fort Myer and the U. S. Naval

Observatory,

V zz 299 709 kilometers zz 1 86 232 miles

(b) From 39 experiments, made in August and September, 1881, across an interval
of 7*4424 kilometers — 4-6245 miles, between Fort Myer and the Washington Monu-
ment,

V zz 299 776 kilometers zz 186 274 miles

(c) From 65 experiments, made in July, August, and September, 1882, across the
above-mentioned interval between Fort Myer and the Washington Monument,

V zz 299 860 kilometers zz 186 326 miles

If these results are to be combined, according to Newcomb we should assign the
weight 2 to (a), 3 to (b), and 6 to (c) ; but he preferred to use (c) alone, on the ground
that it is probably least affected by constant errors.

Miciielson found, from his experiments in October and November, 1882,

V zz 299 853 ± 60 kilometers zz 186322^=37 miles

This result rests upon 23 experiments, made by the revolving-mirror method,
working across an interval of 204935 feet zz 03881 of a mile zz 0*6246 of a kilometer,
at the Case Institute, Cleveland, Ohio. J

Now let us examine Cornu's results a little more closely. His experiments cover
a wide range in the speed of the toothed wheel, and his mean result for each speed is
given in Table II, where V is the velocity of light derived from an experiment in
which i(2w— 1) teeth of the wheel passed during the interval between the departure
and the return of the light, and p is the weight of V. With respect to the experiments

•141, p. 269. f 140, pp. 194, 201 , and 202. % 139, p. 244.

ON THIs SOLAR I'AKAI.l.AX AND ITS RELATED CONSTAN1

3'

made in 1H74, it is to bo observed that the values of V are tin; weighted means of

Cornu's uncorrected values of i(V + v) and \( U -f- a), the weights being taken just as
Cornu gave them.

Table II. — Cornu's experiments on the velocity of light. (130, p. 171 and 131, pp. A. 2(;ii-7.)

Experiments of 1872.

Experiments of 1874.

2» — 1

V

P

211 1

V

r

3
5
7
9
11

13

302-5
297-7
298-2
2988
297-5
300-5

129
2095

4 39'

4 783

924

260

7

9
11

'3
15
17

19
21

23

25
27
29
3'

33
35
37
4'

300 166
300 620
300 050
302 068
J99 960
300 100
300 224
jOO 359
300 500
300490
300304
300304

299 874
299 843
300083

299 55o

300 097

2 511

2 662
2 197

3 150
32946
28880
32 193

1 587

6 250
ij 122
54665

6727
25 047
29 400

6845
45 387

Helmert has pointed out that in the observations of 1874 V seems to diminish
as {in — 1) increases; whence he infers that*

V - V +

y

'.a — 1

(93)

where V is the true velocity of light, and y is a constant depending upon the conditions
under which the experiments were made. If we put

Y-C + x

V' = C + m

(94)

and substitute these values in (93), the observation equations for determining V take
the form

U

211— 1

(95)

and the weighted normal equations will be of the form

0= ' [p]oc+ [2^TI]2/-I>m]

0 = [ 2^7] x + [(-2M - 1?] y ~ [s£J

(96)

*i36, p. 126.

32

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Applying this to the observations made in 1872, with 0 = 2984, the weighted
normal equations are

0 = + 12 582-00^+ 1 724730?/+ 188-200

0 = + i724'73^+ 255-969//+ 63-482

and their general solution is

x = —0001 040912 P + 0-007013 710 Q

y=z + 0007013 710 P — 0051 165396Q

where

P = + 188-200 Q = + 63-482

Whence

x = + 0-249 344 700 ± 0844 7

y — — 1-928 ioi447±5'922 3 (97)

V zz 298 649-3 i 8447 kilometers

with which the residuals in the normal equations are

+ o-ooo 606 + o-ooo 085

the residuals in the weighted observation equations are

— 5i'73 +11 53 +29-62

+ 25-82 —2526 —32-19

and the probable error of an observation of weight unity is ±26 183 kilometers.

Similarly, for the observations made in 1874, with 0 = 300000, the weighted
normal equations are

0 = + 294 402 x+ 1 1 808 y— 51 012 357

0 = + 11 8082; + 546/y— 2455681

and their general solution is

ioox = — 0002 561 621 1 P + 0*055 398 574 2 Q
1002/ = + 0-055 398 574 2 —1-3812204464

where
Whence

P = -5ioi2357

Q- — 2455681

x = — 53'668 960 ± 1 1 1 '8 kilometers

y = + 5 658-249 626 ± 2 595 kilometers (98)

V z= 299 9463 ±111-8 kilometers

with which the residuals in the normal equations are + 5-4 and +o-2; the residuals in
the weighted observation equations are

+ 1 7 009

— 2 251
+ 21 194

— 79091
+ 20395

+ 32517

+ 3420

— 25694

— 12 247

— 25 102

— 16975

— 38 006

+ 20897
+ 43 500
+ 4287

+ 45 423
— 2698

and the probable error of an observation of weight unity is ± 22 080 kilometers.

ON THE SOLAR I XRAI.I.AX AN!) ITS RELAT] D CON I U

33

The values of V in (97) and (98) require to be multiplied b\ 1-000273 in order
to reduce them to what they would have been in a, vacuum; and thus we find

1872. Vzz 298 731 ±845 kilometers = 185624^525 miles
1874. V = 300 028 ± J 12 kilometers = 186 430 i 69-5 miles

which we shall employ instead of the numbers given by Count himself.

Collecting- our results, we now have the following measurements <»f the velocity
of light in vacuo per second of mean solar time, together with their estimated weights:

Kilometers. Miles. Weight.

1849. FlZEAU .515 ;-4 195 935 O

1862. FOUCAULT 298574 185 527 I

1872. CORNU 29S 731 [85624 I

1874. COKNU 300028 186430 2

1879. MlCHELSON 2999IO 186357 3

1881. Voim; and Forbes 301 384 187 273 1

1882. Newcomb 299 860 186 326 6

1882. MlCHELSON 299 853 186 322 3

Fizeau's measurement is rejected on the ground that he himself regarded it as only
a preliminary attempt. The weighted mean of all the other measurements gives

V — 299 835 ± 1 54 kilometers rz 186 3 10 ± 95"6 miles (99)

But if we consider only the four measurements whose weight is greater than
unity, their weighted mean will give

Vrr 299 893 i 23-0 kilometers ■=. 186 347 i 14*3 miles Ooo)

Thus it appears that in either case we arrive at substantially the same value of V,
but with widely different probable errors. We shall adopt the value (100), with a
probable error equal to the difference between (99) and (100), namely,

V — 299 893 i 58 kilometers ='186 347 i 36 miles (101)

18.— MASSES OF THE PLANETS.

We need the mass of the Earth as one of the elements for finding the solar paral-
lax, and the masses of all the other planets are required in computing the luni-solar
precession from the general precession. The following are some of the most note-
worthy determinations of these masses ; others may be found in Houzeau's Vade-
Mecum de l'Astronomie.

Reciprocals of the moss of Mercury.

1782. La Grange, from his hypothetical relation between the densities of the planet-, and their distances

from the Sun. (167,]). 190.) 2025810

1841. Enckk, from the perturbations of the comet which bears his name, during its apparitions in 1819,

1825, 1828, 1835, and 1838, before perhelion. (153, p. 5.) 4 S65 751

185 1. Enckk, from the perturbations of the comet which bears his name,

(<?) From the normal places in l828to 1848, excluding those of 1818, 1822, an< I 1825. (156^1.47.) . 10252900
(i) from the normal places before perhelion in 1828 to 1848, excluding tho . 1822, 1825,

1832, and May, 1842. (156^.49.) 8 234 192

(ir) From the normal places, 1818 to 183.8, without distinction. (156, p. 51.) 3200448

(//) From all normal places without distinction, 1818 to 1848. (156, p. 51.) 3271742

ENCKE thought the mean of (a) and (f) must be near the truth, namely (156, p. 52) 4878172

6987 3

24 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

1877. VonAsten, from perturbations of Encke's Comet, 1818 to 1868. ( i45» P- 98) 7636440

1886. Backlund, from perturbations of Encke's Comet, 1871 to 1881. (146^.37.) 2668700

1S42. Rothman, from the motion of the perhelion of Venus. (184, p. 180.) 3182843

1861. Le Verrier, from the perturbations of Venus. (171^.92.} 5 3IOOO°

1861. Le Verrier, in his tables of Mars. (8, T. 6, pp. 21 and 308.) 4316547

1881. Tisseraxd, upon rediscussing certain of Lie Verrier's equations, found that they were capable of

two solutions. (189, p. 656.)

The one which he regarded as most probable gave 7 100 000

The other gave 3 8o° o00

1882. Newcomb, by estimation, from a consideration of all the data. (177^.468.) 7500000

Reciprocals of the mass of Venus.

1779. La Grange,

From the motion of the Earth's perhelion. (166, p. 115.) 315 517

From the periodic perturbations of the Earth. (166, p. 116.) 341 413

1802. La Place, from the secular diminution of the obliquity of the ecliptic. (7, T. 3, liv. 6, chap. 6, J 21. J 383 13)

1802. Delambre, from the perturbations of the Earth. (7, T. 3, liv. 6, chap. 16.) 356632

1803. Wurm, from observations of the Sun. (191, p. 153.) 326849

1813. BurckhaiIdt, from the pertuabations of the Earth. (151, p. 343) 401 839

1813. Lixdex.-U', from the secular motions of the node and perhelion of Mercury. (173, p. 30.) 349440

1827. Airy, from observations of the Sun made at Greenwich. (142^.30.) 401 211

1842. Rni iimax, from tlie secular motion of the node of Mercury. (184, p. 181.) 362017

1843. Le Verrier, from his equations for correcting the elements of Mercury. (169^.354.) 390000

1853. Hansen and Olufsen, in their tables of the Sun. (40, p. 1.) 40S 134

185S. Le Verrier, in hi> tables of the Sun. (41, p. 102.) 400246

1861. Le Verrier, in his tables of Mars. (8, T. 6, p. 309.) 412 150

1S72. Hn.i., from the motion of the node of Venus. (162, p. 36.) 427 240

1S76. Powai.ky, from observations of the Sun, made at Dorp.it. (181, p. 265.) 396980

1881. Tisseraxd, from the variation of the obliquity of the ecliptic. (189^.658.) 425500

1882. Newcomb, by estimation, from a consideration of various results. (177^.472.) 405000

Reciprocals of the mass of the Earth.

[Values of the Earth's mass obtained from the solar parallax do not come into consideration here.]

1832. Plana, from the parallactic inequality of the Moon. (63, T. 3, p. 20.) (©without j.) 352359

1S63. 1 1 axsex, from the parallactic inequality of the Moon. {222, p. 11.) (©without (J.) 319 455

1872. Le Verrier, from the action of the Earth on the other planets. (172^.169.) (© — ([ •) . . • • 324490

1881. Tisseraxd, from a rediscussion of Le Verrier's equations. (189, p. 658.) (©^ ([ •) 325700

Reciprocals of the mass of Mars.

1782. La Grange, from his hypothetical relation between the densities of the planets and their distances

from the Sun. (167, p. 190.) I 846082

1802. Delambre, from the perturbations of the Earth. (7,T. 3, liv. 6, chap. 16.) 2546320

1813. Kurckhardt, from the perturbations of the Earth. (151, p. 343.) 2680337

1827. Airy, from observations of the Sun made at Greenwich. (142^.30.) 3734602

1853. Hansen and Olufsen, in their tables of the Sun. (40, p. 1.) 3200900

1858. Le Verrier, in his tables of the Sun. (41, p. 102.) 2994790

1876. Powai.ky, from observations of the Sun made at Dorpat. (181, p. 265.) 2876000

1876. Le Verrier, in his tables of Jupiter. (8, T. 12, p. 9.) 2812526

1878. A. Hall, from his own observations of the elongations of the satellites. (158, p. 37.) 3093500^:3295

Reciprocals of the mass of Jupiter.

1726. Newton, from the elongations of the fourth satellite, observed by Pound. (9, lib. 3, prop. 8, cor. 2.) . 1 067

1782. La Grange, from a recomputation of the same observations. (167^.183.) 1067-195

1S02. La Place, from the same observations. (7, T. 3, liv. 6, chap. 6, § 21.) 1067-09

1821. Bouvard, from the perturbations of Saturn. (150, p. ij; compare also 8, T. 12, pp. 67-70.) .... 1070-5

1823. NicoLAi, from the perturbations of Juno. (178, p. 226.) 1053924

1826. ENCKE, from the perturbations of Vesta. (152, p. 267.) 1050-36

1837. Airy, from elongations of the fourth satellite, observed by himself. (143^1.47.) 104677

1842. Bessel, from his heliometer measurements of the elongations of the fourth satellite. (147^.64.) . . 1 047-879 ±0-158

1872. Moller, from the perturbations of Faye's Comet. (175, p. 95.) .... 1 047-788 -|- 0-185

1873. Krueger, from the perturbations of Themis. (165,]). 14.) 1 047558 -J- 0-052

18S1. S< hur, from his heliometer measurements of the elongations of all the four satellites. (186, p. 293.) . 1 047232 -j- 0-246

1888. Haerdtl, from the perturbations of Winni omet. (157, p. 262.) 1 047-175 -J- 0014

ON THE SOLAR PARALLAX AND [TS RELATED CONSTANTS 35

Reciprocals of the moss of Saturn.

1726. Newton, from the elongations of Titan. (9, lib. 3, pro]). S, cor. 2.) 3021

17S2. La Grange, from the elongations of Titan. (167, p. 186.) 3358-40

1802. La Place, from the elongations of Titan. (7, T. 3,liv. 6, chap. 6, | 21.) 3 359-40

1S21. Bouvard, from the perturbations of Jupiter. (150, p. ij.) 3 512

1833. Bessel, from his heliometer measurements of the elongations of Titan. (148^.24.) 3 501-6 -J- 0-78

1S76. LeVerrier, in his tables of Jupiter. (8, T. 12, pp. 9 and 70-72.) 35

1885. A. Hall, from his observations of the elongations of Iaperus. (158^, p. 70.) 1481 0-54

H. STRUVE, from his observations of the elongations of Iapetus and Titan. (187 >2, pp. 1 17-1 18.) . 3498-0 | 1.17

1889. A. Hall, Jr., from his heliometer measurements of the elongations of Titan. (161, p. 146.) .... 3 500-5 ± 1 h

Reciprocals of the mass of Uranus.

1789. WURM, from IIf.rschel's measurements of the elongations of the exterior satellite. ( 190, p. 214.) . . 16 959

1S02. La Place, from the same observations. (7, T. 3, liv. 6, chap. 6, \ 21.) '9504

1821. Bouvard, in his Tables astronomiqa.es. (150, p. ij.) 17 918

183S. Lamont, from his measurements of the elongations of the second and fourth satellites. ( 168, p. 59.) . 24 605

1871. Von ASTEN, from elongations of Obcron and Titania, observed by Lamont, O. Struve, Lassell,

and Marth. (144, p. 21.) 22

1875. Lord Rosse and Dr. CoPELAND, from their observations of Oberon and Titania. (1S2, p. 304.) . . . 24000

1875. Newcomr, from his observations of the elongations of all the four satellite. 1 . 1 ;>•, p. ?6.) 22 540 -j- 50

1885. A. Hall, from his observations of the elongations of Oberon and Titania. (159, p. 33.) 22 682 -j- 27

Reciprocals of the mass of Neptune.

1847. O. Struve, from his own observations of the satellite. (188, p. 815.) 14 494

1848. B. Peirce, from the perturbations of Uranus. (179^.205.) 20000

1849. HlND, from elongations of the satellite, measured by BOND, Lassiii, ami ( ). STRUVE. (l63,p. 203.) 17 900

1850. G. P. Bond, from elongations of the satellite, observed at Cambridge, Mass. (149, p. 3S.) 19400

1854. Hind, from elongations of the satellite, observed by Lassell in 1S52. (164^.47.) 17 135

1S62. Saffok I), from the perturbations of Uranus. (lS5,p. 144.) 20 039 -j- 295

1875. Newtumi'., from his own observations of the elongations of the satellite. (176, p. 63.) 19 380 -L_ 70

1885. A. Hall, from the elongations of the satellite (160, p. 26) :

(a) From his own observations 19 092 -}- 64

(6) From Holden's observations 18279-^ 114

(c) From Lassell and Marth's observations 17 850 -L. 180

The various values given above for the masses of Mercury and Venus differ so
largely among themselves that no trustworthy result can be deduced from them, and
a rediscussion of the original data seems the only satisfactory course. For the masses
of the planets outside the Earth the following numbers will be adopted:

Mass of Mars zz Oo2)

3 093 500

Jupiter = (io3)

1 047-55

Saturn = ( 1 04)

3 501 -6

Uranus — - O05)

22 600

Neptune =— — — — (106)

r 18780

Number (102) is Professor Hall's value. For the mass of Jupiter the arithmeti-
cal mean of the values given by Bessel, Moller, Kruegek, Schur, and Haerdtl
is 1 : 1 047*522, while the mean of the two values given by Bessel and Schur is
1 : 1 047-555. Number (103) is sensibly the latter value. According to II. Struve,*

* 187^, pp. nS-119.

■>6 ON THE SOLAR PARALLAX ANT) ITS RELATED CONSTANTS.

the revised value of Bessel's mass of Saturn is 1:3 5025. The mean of that, and
the value found by Mr. A. Hall, Jr., is 1:3 5015, which differs so little from Ves-
sel's own value that the latter has been retained in (104). Number (105) is very
nearlv the arithmetical mean of the values given by Newcomb and Hall, and (106)
is almost the arithmetical mean of the values given by Bond, Newcomb, and Hall,
the latter including the observations by Holden, Lassell, and Marth.

We now proceed to determine the masses of Mercury, Venus, and the Earth from
the perturbations, both periodic and secular, of these planets and of Mars. To facili-
tate the discussion of the motions of the nodes of Mercury and Venus, let

Q, =. longitude of the planet's ascending node,
Qo = approximate value of &, such that Q, rr Q,0 + -dQ,,
m rr mass of the planet,

m0 z= approximate value of m, such that m rr m0( 1 -f- v),
i z=. inclination of the planet's orbit to the plane of the ecliptic,
tzz time in Julian years of 365^ days, counted from a specified epoch,
(/A© rr correction to the assumed value of the Sun's longitude,
rffi© = correction to the assumed value of the Sun's latitude,
di — correction to the assumed value of ?',
dp and 6q zz certain coefficients whose numerical values are given by Le Verrier in
the Annales of the Paris Observatory, Tome 2, pp. 100-102.
Further, let symbols relating to the different planets be distinguished by superior
Roman numerals, in the usual way, those relating to Mercury being without any
numeral, while those which relate to Venus are marked ', those which relate to the
Earth u, and so on to Neptune, symbols relating to which will be distinguished by
the numeral vu.

In order to make use of Professor Newcomb's investigation of the longitude of the
node of Mercury, let us put

0, — Qo + ^Q, (107)

where the value of &0 is that given by Le Verrier for the epoch 1 850*0, in his tables
of Mercury,* namely,

£0 = 46° 33' 0875" + 42-643"* + 0-000083 5"t2 (108)

From a discussion of 23 transits of Mercury, occurring between the years 1677
and 1 88 1, Newcomb found for the epoch i820'ot

N — N0 + Wt - - o- 1 6" ± o- 1 8" + (0-28" ± 0-42")*

Whence, for the epoch 18500

N==-o-i6//±o-i8" + (o-28"±o-42")-i?52zl^?+(o-28//±o-42,/)<

— _ 0-076" ± 0-220" + (0-28" ± O.42")*

Newcomb wrote

N = (£0 — <W')8in*

•170, p. 107. 1 1 77. P- 460.

ON THE SOI \K PARALLAX AND Ms RELATED I NTS.

37

but, as 6V can not be evalued, when the unit of time is changed from a century to a
year this may be regarded as equivalent to

Sill I IOO Sill /

which, with i— 7" 00' 077", and the above numerical value of N, gives

^£ = -0-623" ± 1 -805" + (0-023" ±0-034";/ (109)

By substituting (108) and (109) in (107), we obtain definitively for Mercury
£ = 46° 33' o8-i3"db r8o'/ + (42-666"_t-o-034")£ + o-oooo835//*2 (,IO)

where t is counted in Julian years of 365 \ days, from 18500, Paris mean time.
Passing next to Venus, Encke found from the transit of 1 761*

Epoch, 1 76 1, June 5d 1 71' 30"1 Paris mean time,

O / //

Longitude of Sun 75 35 49'6

+ o-6
3 23 26

Latitude of Sun

Inclination of Venus's orbit . . .
Q1 = 74° 3i' 54,4°" + M©- 16-882^0— 10-919^ + 0-313^
And from the transit of r 769*

Epoch, 1769, June 3d ioh iom Paris mean time.

Longitude of Sun ....
Latitude of Sun ....
Inclination of Venus's orbit .

o 1 //

73 27 i3'8
o-o

3 23 26

Q,1 =74° 36' o8-6o" + ^A©— 16-882^©+ 10-45 ufyj — 0-339^

In order to reduce these results to conformity with the positions of the Sun given
by the Tables du Soleil of Hansen and Olufsen, and to the inclination of the orbit of
Venus given by Hill's tables of that planet, we take from the latter work, Introduc-
tion, pp. 2 and 23 :

Transit of 1761. Lransil ol 1769.

Longitude of Sun

0 / // 0 / //
75 35 5205 73 27 14-25
f 053 + 004
3 23 3164 3 23 31-94

{i

*202, pp. IO5-IO7.

38 ON niK SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Whence, by comparison with Encke's results, and from Hill's tables of Venus,

<Taq

43©

di>

Perturbations of Venus in latitude

Encke's observed Q,' . . .
Correction for i/'/.Q . . . .
Correction for </3q . . . .

Correction for t/i>

Correction for perturbations

Corrected Q'1 . . . .

Transit of 1 76 1.

//

+ 2-45
— 0-07

+ 5-64
-|- 0-060

Transit of 1769.

//
+ o-4S
+ 004

+ 5'94
+ 0-084

o / //

74 3i 5446
+ 245
-f 1-182
+ 1765
+ 1013

74 32 00 87

o / //

74 36 08-60

+ °-45

— 0-675

— 2-014
+ I-4I8

74 36 07-78

The provisional expression for the longitude of the node of Venus, employed by
Hill in the construction of his tables of that planet, was

Q,i=75° 19' 52-3" 4- 32-293 i"£ + o-oooi 51 *2

where t is reckoned in Julian years from the epoch 1850*0. To this, from the meridian
observations made at Greenwich, Paris, and Washington during the years 1836 to 1871,
he found the correction*

sin 1 C059 18

which belongs to the epoch 1855, January 00, Washington mean time. Whence, for
that epoch,

81 = 75° 22' 3578"

From the 1475 photographs of the last transit of Venus, reduced by the United
States Transit of Venus Commission, we have for the epoch 1882, December 6d 5b o'",
Greenwich mean time,

#=75° 37' 33'9ii" +^A 0-16-868^©- 0-321 di{

This result already depends upon the position of the Sun given in the Tables du
Soleil of Hansen and Olufsen, and it requires correction only for the perturbations of
Venus in latitude. According to Hill's tables of Venus, these perturbations amount
to -f- 0-046", and the corrected result is, therefore,

Si = 75° 37' 34W

*l62, Introduction, p. 36.

ON [TIE SOLAR PARALLAX AND US RELAT1 U CONSTANTS.

From Hill's tables of Venus, Introduction, p. 2, we have

39

Gs = 75° *9' 53,io" + 32'5i5o"/ + o-oooi5i//<2

where

£=r(I)ate in Washington mean time)— 1850-0
= (I)ate in Greenwich mean time) — i850'000 586

Computing the values of £l for the dates of observation by this expression, and
collecting our observed results, we have

Julian date.

Computed Q'

Observed &'•

C — O

0 / //

0 / //

//

1761-398967

74 31 54'49

74 32 00-87

-6-38

1769392654

74 36 J4'20

74 36 °778

+ 642

1854-966 363

75 22 35-66

75 22 35-78

— 0-12

1882-897 900

75 37 4401

75 37 34-69

+ 932

In order to form the observation equations for the determination of the corrected
expression for &\ let

0=75° 19' 53-10" +£ + (32-51 50" + //)<+o-oooi 51 *2

— (observed Q? at the time t)

(in)

and then, by putting C for the computed, and O for the corresponding observed value
of Q,', we shall have

o = (C — 0)-f x + ty

The scale of the Julian dates given above is such that 1849-967042 corresponds
to 18500 Washington mean time. Bearing that in mind, the following observation
equations result from the values of (C — O) :

ozr — 6-38 + 3: — 88-568?/
0 = 4-6-42 +£ — 80-574//
0 = — OT2+.r + 4-999//
0 = + 932+.r +32-93 I//

Weight 1
Weight 1
Weight 1
Weight 4

The probable error of the observed right ascension of Venus was +0470'' in
1761, ±0-496" in 1769, and only ±0038" in 1882 Therefore, according to theory
the weight of the last observation equation would be more than 150 times that of either
of the first two. Nevertheless, on account of the possible existence of constant errors,
it has been thought prudent to give it a weight of only four.

(112)

4o ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Having- regard to the adopted weights, the normal equations are
o = + 37-200"+ 7'ooox— 32-4193/

0 = T 1274-847" — 32-4I9Z+ 18699-253//

and the general solution is

x = — 0-144013 468 P — o-ooo 249 677 Q
y = — o-ooo 249 677P — 0-000053 91 1 Q

in which

P = + 37-200" Q = + 1274-847"

From the general solution

x — — 5675 601" ± i'973" V = — 0-078016 2" ±0*038 2"

with which values the residuals in the normal equations are respectively 00000" and
— 00004", while those in the weighted observation equations are respectively — 5'i46,"
+ 7030", —6- 1 86", and +2-151".

If it is desired to use only the data afforded by transits of Venus, we may take
J (1 76 1 + 1769) and 1882. In that way dp, the unknown correction to the adopted
value of the Sun's semi-diameter, is sensibly eliminated from the observations of 1761
and 1769, and our observation equations become

o = + 0-02" +# — 84-571?/
o = + 9-32" +£ + 32 93 1 #

Whence

x = — 6-714"

y — — 0-079 147

The difference between this result and that from the normal equations ^112) is
less than the probable error of either.

Reverting to the values of x and y yielded by the equations (112), and substi-
tuting them in (in), we obtain definitively for Venus

S1 =75° 19' 47-42" ±r97" + (32-4370" ±00382")* + 0-000151"^ (113)

which belongs to the epoch 18500 Washington mean time.
If we put with Le Verrier*

p — tan i sin Q, q = tan i cos &

then, regarding all the quantities as variable, and differentiating

dp = ,. di + tan i cos Q-dQ

COS' 1

dq = - , . cji _ tan i sin 0,'dQ,
cos- 1

*8, T. 2, p. 26.

ON THE SOLAR PARALLAX AND lis RELAT] I- CONSTAN 41

whence, by eliminating di, we find for the theoretical motion of the node on the fixed

ecliptic

dQ rz coti cos Q-dp — cot I sin Q-t/q

To find the theoretical motion on the movable ecliptic, it is only necessary to
substitute dp — dp" and dq — dqli for dp and dq. In that way wo obtain lor Mercury
and Venus

dQ, — cot i cos Q {dp — dp") — cot i sin Q (dq — dqa) (114)

dQ} = cot il cos S1 (dp1 — dp") — cot i1 sin Q' (dq[ — dqli) (115)

From equations (no) and (113), and from Le Vereier's tables of Mercury and
Hill's tables of Venus, we have for the epoch 18500

Q =46° 33' 08-13" i -70 00' 0771

a' =75° 19' 47-42" i;=3° 23' 35-oi

By the substitution of these values, and after replacing dp, dp\ dp", dq, dq\ and
dq" by the numerical values of dp, dp\ dpu, dq, dq\ and dq*, from the Annates of the
Paris Observatory, T. 2, pp. 100-101, equations (114) and (115) become

(l?=— 7-60689"— 0-065 oo'V —4-102 14' V1 — 0-923 46' VH
' — o-i 12 38'Vm— 2-283 53'Viv— 011704"^

— o-ooi 84'Vvi

dO}
,. =- i7,367 93" + 0-11193" v - 5-038 55'V -6-71749"^

(Il6)

— 0-22172 Km— 5-22275 f"'— 0-27355

— 0-004 i8"v

Putting ^ for the general precession, the observation equations for determining
the masses of Mercury, Venus, and the Earth will be of the form

o = theoretical -p -4- fa — observed — p (117)

dt ill

As the masses of all the planets outside the Earth's orbit are here regarded as
known, the numerical values of their vs must be substituted in the observation equa-
tions. To find them, we have the relation

m = »i0( 1 -f- v)

where w0 is the value employed in the observation equations for the mass of any planet.
or, in other words, it is the value employed by Le Verrier in forming the quantities
dp, dq, etc., and m is the adopted mass of the same planet. The numerical values of
1 -^ w0, 1 •— m and v are given for each planet in Table III.

42

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Table III.

Planet.

I -r- m%

Le Verrier.

I -f- m
Adopted.

Factor

V

Earth

3 ooo ooo
401 847

354 936

2 680 337

1 050

3 512
24 000
14 400

Indeterminate . .

Indeterminate . .

Indeterminate . .

3 093 5O0

1 04755

3 501-6
22 600
18780

— 0-I33 558
+ 0-002 339
-)- 0-002 97O

-)- 0-061 947

— 0233 227

Taking the observed values of the motions of the nodes from equations (1 10) and
(113), the theoretical values from the equations (116), and the factors for the correc-
tions of the masses from Table III, the formation of the absolute terms of the observa-
tion equations will be as follows :

Mercury.

Venus.

Provisional motion of node .
Correction for mass of Mars .
Correction for mass of Jupiter
Correction for mass of Saturn

rr

— 7-606 89
-j- 0015 01

— 0005 34

— OOOO 35
+ 50237 19

rr

— I7-367 93

— 0-029 61

— 0-012 21

— OOOO 8l
+ 50237 19

Theoretical dQ,/dt ....

+ 42-639 62
-j- 42666 O

+ 32885 85
+ 32-437 ©

Observed dQ/dt

(C — 0) = absolute term . .

— 0-026 4

+ 0-448 9

The observation equations are therefore

// // // // //

Ozz — OO659 v — 4-1021 v1 — 09235 vn — 0*0264 zh 0*034

O = + O" I I 1 9 v — 5-0386 vx — 67 1 75 v" + 0*4489 ± 0-038

whence, leaving vh indeterminate,

(118)

v — + 28-951 626^— 2-495 943 ±0-427
v% — — 0-690 234 vil -f- 0-033 66 1 ± 0005 75

If we assume vn zz + 0*066 631, then

v — — 0-566 867 ± 0-427
vl = — coi 2 330 ± 0005 75

In the Annales of the Paris Observatory, T. 6, pp 286 and 307, Le
has given four equations which may be used for determining the masses of

Vekrier
Mercury,

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS 43

Venus, and the Earth. They are derived respectively from tin- secular motions of the
perihelion, eccentricity, inclination, and node of .Mars; and, alter multiplication by 100,
they are as follows:

(119)

o = + o-i4v-f 4*66 k'+ 16-36 ku+ 1306 k1*— 2*353
o = + ooSk + 0*69 v[ -\- 2-o6k"-|- iS-2 k1v — 11 15
o = — oiok+ 1222 »''+ C03 y" — 131 1 vlv + 0565
o = — 0*69 v — 25-60^— 6-821'"— 37 15 viv — 0-577

Regarding v, v\ and v" as unknown, the normal equations are

// // // " . "

0 = + 05121 k + 1 7-1496 vl-\- 7-i58oKn+ 466845 y'v — 0-0770

0 = + 17-1496^ + 826-8801 ^+252-6176^'+ 141 1-9898 k'v 4- 9-9411 (120)

0 = + 7-1580^ + 252-6 1 76 ^ + 318-4065^ + 24270777 i'iv — 36-8399

and the general solution is

v z=—7-o84 439P +0129 718Q + 0-056 347 7 Ii
\ov' — + 1-297 176P — 0-039 714Q + 0002 347 2R
100 y" — + 5634 770P + 0-023 472 Q — o 459 359 4^
in which

P = + 46-6845 Kiv— 00770

Q = + 1411-9898^+ 9'94n
R — + 242 7-0777 vw — 36-8399

By substituting the value of viY from Table III these quantities become
P = + o-032i Q=:+ 13-2423 R = — 31-1654

and them from the general solution,

v — — 0265 744 ± 1-511
v1 — — o 055 742 ± 0-035 76
v" — + 014S07S + 0-03847

With these values of the ks, the residuals in the normal equations (120) are
+ 0000001" +0-000055" —0000008"

the residuals in the observation equations (119s) are

+ 00779" —0-8272" — 0-1159" —00634"

and the probable error of any one of the observation equations (119) is =1=0-5675".
We have next to deal with the following- group of equations :

0=: o'oo>' + 29/-5 ^ + 225-3 v"— 18-59

o = — 2 7-39" -46-33 yi~ 51'59>/ii— l8'°2

o=:+i4-3 " + 25-5 v* + 277 "" + 17

o-+ 7-8 k+ 9-2 V + 153 ^+ 37 C121)

O — — 0-53^ + 24-6 v1 + 32.8 va— 186

O — — I-24V + 40-4 VJ + 54-0 Vil— 3-28

o — + 0-53^ + 28-88^ o-o v"+ 174

44

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

When Mais made its near approach to tf1 Aquarii on October i, 1672, the position
of the planet was compared with that of the star by Richer at Cayenne, by Picard near
Beaufort, and by Roemer at Paris. From a very careful discussion of these compari-
sons, Le Verrier derived the equation which he has given in the Comptes Rendus,
T. 75, p. 169, namely,

o = + 29-5" vx + 225-3" v" + l39%" yiv—2i -86"

By substituting in it the value of vivfroni Table III, the first equation of the group
(121) results.

The second equation of the group (121) is from the Annales of the Paris Observ-
atory, T. 6, p. 72>'i ana" is based upon the longitude of Venus deduced from Horrox's
observation of the transit of that planet in December, 1639. The third and fourth
equations are from the Annales, T. 6, p. 76; the former being derived from the lon-
gitudes of Venus obtained from Bradley's meridian observations, and the latter from
the longitudes obtained from the meridian observations made between the years 1 766
and 1830. The fifth and sixth equations are from the Annales, T. 6, p. 90; the former
being derived from the latitudes of Venus resulting from the observations of the tran-
sits in 1 76 1 and 1769, and the latter from the latitudes resulting from the Greenwich
meridian observations made between the years 1751 and 1830.

In the Annales of the Paris Observatory, T. 4, p. 52, Le Verrier has given the
equation

oz + 0-53" v + 28-88' V + 0-83" vm + 1 -8 1"

(122)

which he has derived from the secular diminution of the obliquity of the ecliptic. By
adding the term + i6-oi'Vv, from p. 51 of the Annales, and substituting the values
of vhi and v** from Table III, the last equation of the group (121) results. In the
Annales, T. 5, p. 100, and T. 6, p. 91, and in the Comptes Rendus, T. 75, p. 168, Le
Verrier has given equation (122) in the form

o = + 0-53" v + 28-88" v1 + 0-75" vm + 1-72"

(123)

but the difference arises solely from the circumstance that in (122) the assumed mass
of Mars is 1 : 2 680337, while in (123) it is 1:2 994 790. For further comments on
the equations (121), Tisserand's paper in the Comptes Rendus may be consulted.*
From the group of observation equations (121) the following normals result:

o
o
o

+ 1017-641' 4- 1 657-55 ^+ 1844-16^+ 552-713
+ l657'55,/+ 6822-98 k'+ 12 872-101'"+ 235-835
+ 1 844-16 v -\- 12 872- 10^+ 58 414-841'" — 3 393-103

(124)

189.

ON III1. SOLAR PARALLAX VND ITS RELATED CONS! VNTS. 45

and their general solution is

1 000 v — — 1 764489 702 P + 0553 787 566Q —0066325 742 R
1 000 ^ = + 0-553 7%7 566P — 0-424651 923Q +0-076091 780 I {
1 000 k" — — 0-066 325 742 P + 0076091 780Q —0031 792 362 R

where

P=r + 552713 Q = + 235-835 R = - 3 393'I03

We therefore have

v — — 0619 603 8 + 0-096 92

vl = — 0-052 2494 + OO4755
v" = + 0-089 1 60 8 + orj 1 3 o 1

with which the residuals in the normal equations (124) are

+ o-ooo 2" + o-ooo 8" + 0-002 8"

The residuals in tne observation equations (121) are

— 0-043" —6-022" +0-107" —0-097"

— 3*228" —0-250" +0-192"

and the probable error of any one of the observation equations (121) is + 2-307".

The equations in group (125) are from the Annales of the Paris Observatory, T.
4i P- 95> and depend entirely upon observations of the Sun. The first equation of
the group has been deduced from the differences of the maximum values of the equation
of the center determined at two epochs fifty years apart. The second equation arises
from the observed motion of the Earth's perigee; and the remaining equations are
based upon the periodic perturbations of the Earth, the third and fifth arising from the
action of Venus, while the fourth and sixth arise from the action of Mars.

// // // //

0 = — 0*23 v + roi y' — C65 vm — o*2i

o = — o 43 v + 5-97 vi + 1 93 vni — o 44

0 = — 004^+ S-oo^1 — 013 vm — 001 (12O

O rr + 002 v + r -07 k1 + 4-00 v1" + 0-48

O = — 002 v + 800 v1 — 0'17 v>u — 009

0 = o*oo v + o-6 1 vi + 4-00 vlu + 0-3 5

In accordance with Le Verrier's estimate of the relative accuracy of these equa-
tions,* the second will be given a weight of !,. It then becomes

o = — o-i4"v+ 1 -99" k1 + 0-64" vUi — 0-15" (126)

*4'> PI>- 95-96-

46 llN' I HI SOLAR PARALLAX AND [TS RELATED CONSTANTS.

and, regarding v'" as known, the weighted normal equations are

o = + 0-0749" v — 0-9695" v[ + o- 1 485" Kin + o-oS 1 1 "
o = -o-9695'V+ 134-4972' V' + 4-937i" ^-0-5835"

The general solution is

v zz — 14-725 041 P — 0*106 143 Q

(127)

where

v* — — o' 1 06 143 P — 0008 200 Q

P = + 0-1485 ^" + 0-081 1
Q = + 4-9371 ^"-0-5835

Hence

(128)

v zz — 2-710 707 v'" — 1*132266
vl =1 — 0-056 246 v™ — 0-003 824

and the residuals in the normal equations (127) are

— o-ooo 00" vm + o*ooo 00"
+ 0*000 20" vm — o-ooo 09"

The substitution in (128) of the value of vm from Table III gives

v — — 0*770 229 + 0-317
v* = + 0*003 688 + 0-007 49

with which the residuals in the weighted observation equations (125) and (126) are

+ 0-058" — 0-120" +0-068" —0-065" — 0-022" —0*l82"

and the probable error of any one of these equations is +0082 76".

Collecting our results, from the groups of equations ( 1 1 8), ( 1 1 9), ( 1 2 1 ), and (125),
the values of v are

— 0-566 867 ±0-427

— 0-265 744+1-511

— o-6 1 9 604 ± 0-096 92

— 0-770 229 ±0-317

and the values of vl are

— 0-012330 ±0005 75

— 0055 742 ±0035 76

— 0-052 249 ±0-047 55
+ 0-003 688 ± 0-007 49

while from the groups of equations (119) and (121) the values of v" are

+ 0*148078 ±0*03847
+ 0089 161 +0013 01

ON THF, SOLAR PARALLAX AND ITS RELATED C0NS1 WIS.

47

Instead of attempting to deduce final values from these results, it will be better
to reduce the various groups of equations to ;i uniform standard of weight, and then
to solve them all simultaneously. The data for that purpose arc as follows:

Equations.

Probable error.

y/ Weight

Group (118) . .
Group (119) . .
Group (121) . .
Group (125) . .

± 0036

±0-507

1 2-307
± 00828

io-ooo

0635

01 56
4-35

The weight of the equations in group (1 18) is arbitrarily assumed to be 100, and
then the weights for the other groups follow from the probable errors given in the
second column. By the application of these weights^ each to its own group of equa-
tions, the subjoined system of weighted observation equations is obtained:

0 = — 0*659 v —
0 = 4 n 19 v —
o zz + 0*089 v +
o z= + 0-05 1 v +
Or — 0*064 v 4
0=3 — 0*438 v —
o zz o*ooo v 4

0-= — 4*273^ —
0 = 4- 2*23 1 v 4-
Ozr + i'2iyv-\-
ozz — 0*083 v 4-
Or — 0*193 v -\-
Orr + 0*083 y +
O zz — 1 *000 v +
o — — 0609 v -\-

Ozz — 0*174 V -\-

o — + 0087-^4-

ozz — 0*087^4-
o = o*ooo v 4-

The resulting normal equations

41*021 v}

9-235 k" — 0*264

50-386 v' -

67 1 75 ""44-489

2959^ +

10*389 v" — 1*300

0-438 v1 +

1*308 y" — o'6S 1

7*760 v> 4-

o-o 1 9 v" 4-0*339

16*256^' —

4-33i ^ — 0422

4*602 vx 4-

35'147,/ii— 2-900

7*227 v1 —

8*048^—2*811

3*978 k* +

4*321 1^4- 0*265

1-435^-4-

2-387^ + 0*577

3*838^ +

5'I 17 vil — 0*290

6302 vi -\-

8424 vA — 0-512

4-505^

O'OOO vil 4- 0*27 1

4"394*'i

o'ooo^" — 0*535

8-656 V1

O'OOOK" — 1*02 2

34*800^-

0*000^-4-0*030

4'654 ^

o-oook*1 — 0*235

34-800 k-

o'ooo -'" — 029 1

2-654 vl

0*000 k" — o* 800

are

(129)

o — + 280773 v — 0*4420 v* — 2 1 *3 1 20 v" 4- 19-8172
o — — 0*4420 v 4- 7265*9429 v- 4- 41 78*6252 yVx — 227*7718
0= — 21*3120^4- 4178*6252 yi-\- 61477642 yli — 394*2509

and their general solution is

iook zz — 3*576 684 283 P 4001 1 349 517Q — 0*020 113 276R
iook* z= + o*oi 1 349 5 1 7P — 0022 63 1 1 15Q +0*015 421 676R
ioo^zz — 0*020 1 13 276 1* 4 0015 421 676 Q — 0*026 81 7 889 R

(130)

48 ()N THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

where

P = + 19-8172 Qrr — 2277718 Rz= — 3942509

We therefore have

v — — 0-655352905^0-06580
v{ — — 0*007 003 642 rb 0-005 235
v" = + o*o66 6 1 7 652 i 0-005 698

with which the residuals in the normal equations (130) are

+ 0'OOOOOl" +0-000 002" + 0-000 002"

The residuals in the weighted observation equations (129) are

II

— 0*1600

— 0-5908

+ 0-1850

— 03664

— 0-4964

+ 0-0895

— 0-6868

— 09370

-06835

— 0-6303

— 0-0715

— 0-0998

+ 0-327*

+ 0-0785

— 03246

— 0-3097

+ 0-1316

-0-4778
— 08186

and the probable error of any one of the weighted observation equations (129) is
±0-3479".

As a final check on the solution, we have the relation

[m«] + [aii] v + [bii] yi + [c»] v" — [vv~] =. o

which is satisfied thus :

//
\wn\ +41-91363

[an\v — 1 2-987 26

[&«>> + 1-59523

[c»]ku — 26-26407

Sum + 4*2 57 53
[w] + 4-257 3i

Check + 000022

It may be remarked that the weight factors were so chosen as to give a probable
error of +: 0-360" for each of the observation equations in the group (129), and the
fact that this probable error comes out ±0-348" seems to indicate that the relative
weights were sufficiently exact. The 9th and 19th observation equations give the
largest residuals, and perhaps it might have been better to omit them, but it is not
likely that their retention can have sensibly affected the corrections to the masses.

ON THE SOLAR PARALLAX \KD ITS RELATED I NTS.

49

From the solution of the group of equations (129) we now have as the definitive
result of this investigation

Mass of Mercury = °'^ 6^ ± °-°65 ?o =

3000000 8704559^1724742

Mass of Venus = °'992 996 ±0-005 235 = 1 , .

401 847 404681 ± 2 134 v ° '

Mass of Earth = ro666 ! 8 ± °°°5 69» = J_

354 936 332 768± 1 77«

Or, expressed decimally,

Mass of Mercury = o-ooo 000 1 14 882 ± o-ooo 000 02 1 933

Mass of Venus = 0-000002 471 082 -4- o'ooo 000 o 1 3 027 (l32)

Mass of Earth = cooo 003 005 097 ± 0000 000 01 6 056

With respect to the data employed in these determinations, transits of Mercury
have been used down to 1882, transits of Venus to 1883, meridian observations of
Mercury to 1842, meridian observations of Venus to 1871, meridian observations of
the Sun to 1850, and meridian observations of Mars to 1858. Since these dates there
have accumulated 47 years of meridian observations upon Mercury, 18 years upon
Venus, 39 years upon the Sun, and 3 1 years upon Mars ; but to utilize them exhaust-
ively for determining' the masses of the three interior planets would necessitate an
amount of labor almost equivalent to computing new tables of Mercury, Venus, the
Sun, and Mars.

Some explanation seems desirable respecting the method of computing the prob-
able errors of the planetary masses in (131) and (132). The expressions for these
masses are of three forms, which may be written

a ± b a , b , >.

« = — = -±- (133)

m = e±/ (134)

™ = — J-T (J35)

Whichever of these forms is employed, it is clear that when the probable error is
added to, and subtracted from, the most probable value of the mass, the resulting limit-
ing values should be the same. That condition is manifestly fulfilled by the forms
(133) and (134), and in order that it maybe fulfilled by the form (135) we must
have

(a + b)/c=i/(g + h) ^^

and also {a — b)/c— 1 / {g — h)

The form (135) is usually derived from (133) by the binomial theorem, thus

1

a\ aJ a a
6987 4

bx~1~c^cb , A 037)

5o

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

all terms of the expansion beyond the second being neglected; and for that reason the
formula so obtained is sufficiently exact only when b is small compared with a. To
obtain a more general expression, we remark that as the maximum and minimum
values of (133) are

a + b

and

a — b

we may write

2I1-

2cb

a-\-b a — b a2 — b2

and therefore, very approximately

«±6_ 1

c c _,_ cb

a^

(138)

As c/a is not precisely equal to

2\a-{-b a — by

formula (138) does not rigorously fulfill the conditions (136), but it is far more exact
than (137) when b is not small relatively to a, and as b diminishes the results given
by (137) and (138) tend to become identical.

By reasoning similar to that employed in deducing formula (138), it is easy to
obtain

c±/=.

/

e ^ e2 -P

039)

which is required in passing from form (134) to (135). Also, for passing from (135)
to (134)

1 _ 1 h

y±h 9 g2 — h2

(140)

But if h has been derived through the forms (138) or (139), as is usually the case,
then according to (133) the probable error is dh^/c, and by expressing that quantity
in terms of g and h we find, with all needful accuracy

1 _ lw h2

g±h~rjg^ l J2)

(141)

ON THE SOLAR PARALLAX AND ITS RELATE! J CONSTANTS. 51

19.— TRIGONOMETRICAL DETERMINATIONS OF THE SOLAR PARALLAX.

Observations of Mars, when in opposition to the Sun, and at its least distance
from the Earth, constitute one of the oldest trigonometrical methods of determining
the solar parallax. There are two ways of making the observations. Either the
planet is observed on or near the meridian, at two stations situated respectively, in
the northern and southern hemispheres; or it is observed soon after rising, and just
before setting, at a single station. The first method will be termed the meridian
method, the second the diurnal method. In the meridian method the observations
may be made either with a transit circle, or with a micrometer attached to an equa-
torial telescope. In the diurnal method they may be made either with an equatorial
telescope, or with a heliometer.

The values of the solar parallax resulting from some of the most noteworthy
attempts by the meridian method are as follows :

//

1672. J. D. Cassini (196, p. 114) 95

1751. La Caille (Ephemerides des mouvements celestes depuis 1765 jusqu'cn 1774. Paris. Introduction, p. 1) . . 10-38

1835. Henderson (224, p. 103) 9-028

1836. Taylor (265, p. 71) 9253

1856. Gili.iss and Gould (216, p. eclxxxviij) 8-495

1863. WlNNECKE (269, p. 264) 8-964

1865. E. J. Stone (252, p. 97) 8943

1865. Asaph Hall (217, p. lxiv) 8-842

1867. Newcomb (232, p. 22) 8855

1879. Downing (198, p. 127) 8-960

1881. Eastman (200, p. 41) 8-953

1882. E. J. Stone (264, p. 300) 8-95

The following are some of the results from the diurnal method :
* //

1672. J. D. Cassini (196, p. 107) I0-2

1672. Flamstead (209) • IO

1719. Pound and Bradley (219, p. 114, and 243, p. 11 11) IO'S

1857. W. C. Bond (195, p. 53) 8-605

1877. Maxwell Hall (218, p. 121) 8789

1879. Gill (214, p. 163) 8-78

Owing to the comparative nearness of the asteroids, and their small, well-defined
disks, it has been thought that the solar parallax might be accurately derived from
observations made upon them in the manner just described for Mars. Several attempts
in that direction are now in progress, but the following are believed to be all the results
hitherto published :

//

1875. Gai.ie, from Flora (211, p. 7, and 213, p. 67) SS73

1877. Lord Lindsay and Dr. Gill, from Juno (230, p. 211)

The same method has also been applied to Mercury and Venus, but there are
great difficulties in the way of obtaining satisfactory results from these planets.

Transits of Venus. — Until comparatively recently, astronomers have believed that
transits of Venus furnish by far the most accurate means of determining the solar

52

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

parallax. Such transits have been observed by three different methods, namely: (1)
by noting the times of contact between the limbs of Venus and the Sun ; (2) by
observing the position of Venus upon the Sun's disk with a heliometer ; (3) by photo-
graphing the Sun with Venus upon its disk, and subsequently measuring the photo-
graphs.

Contact Observations. — The following are some of the results for solar parallax
obtained by different astronomers from contact observations of the transits of Venus
in 1 761, 1769, 1874, and ♦1882:

Transit of 1761.

//

1763. Hornsby (225, p. 494) 973

1763. Short (250, p. 340) 856

I765. PlNGRE (239, p. 32) IO-IO

1767. Planman (242, p. 127) 849

Transit of 1769.

1769. Euler(203^, p. 518) 8-So

1771. Hornsby (226, p. 579) 878

1771. La Lande (227, p. 798) 862

1771. Maskelyne (12, vol. 1, p. 413) 8723

1772. Lexell (229^, pp. 661 and 672) 863

1772. PlNGRE (24O, p. 419) 880

1772. Planman (5, p. 407) 843

1786. Du Sejour (199, p. 486) 8-851

1814. Delambre (3, T. 1, p. xliv) 8552

1815. Ferrer (208, p. 286) 858

1865. Powalky (244, p. 22) 8832

1868. E. J. Stone (256, p. 264) 891

Transits o/1761 and 1769.
1835. Encke (203, p. 309) 8571

Transit of 1874. ,

1877. Airy, from British observations (193, p. 16) 8754

1878. E. J. Stone, from British observations (261, p. 294) 8S84

1878. Tupman, from British observations (267, p. 455) 8846

1881. Puiseux, from French observations (247, p. 487) 893

1881. E. J. Stone, from French observations (263, p. 328) 8-88

Transit of 1882.

1SS7. E. J. Stone, from British observations (251, p. 7) 8832

1887. Cruls, from Brazilian observations (197, p. 1237) 8808

The large differences in the parallaxes obtained by different astronomers from the
same observations are due to the circumstance that, as the instants of contact are ren-
dered uncertain by the intervention of various disturbing phenomena, many of the
observers record two or three different times, corresponding to as many different
phases which they endeavor to describe, and thus the resulting parallaxes are influ-
enced to a certain extent by the interpretation put upon these descriptions. The
interior contacts give better results than the exterior ones, but in any case the prob-
able error is large. From 61 selected observations of interior contacts of the transit
of 1874, discussed by Colonel Tupman,* the present writer found the probable error

^267, twenty on p. 450 and forty-one on p. 453.

*.

ON THE SOLAR PARALLAX AND rTS RELATED CONSTANTS. 53

of an observed time of contact to be ±4'59", which corresponds to a probable error -I"

±0-15" in the distance between the centers of the Sun and Venus. Actual errors of

from 20 to 30 seconds in the observed times of contacts arc by no means uncommon.

Observations with Heliometers. — A few heliometers were used in observing the

transits of 1874 and 1882, but until the resulting values of the solar parallax are
published the accuracy of their work can not be satisfactorily estimated.

Photographic Observations. — For observing the transit of 1874 photography was
extensively employed by the English, French, German, and United States parti<
but the photoheliographs used by the English and Germans differed radically from
those used by the French and Americans. In the subsequent measurement of the
pictures the English and Germans failed to obtain satisfactory results, while the
Americans, and apparently the French also, succeeded completely: and thus it came
about that no photographs of the transit of 1882 were attempted either by the Eng-
lish or by the Germans, while the Americans and French took many hundreds. So
far as known, the following are the values of the solar parallax yielded by the pho-
tographs :

Transit of 1874.

1881. Todd, from the United States photographs (266, p. 493) 8-883^0034

1885. Obrecht, from the French daguerreotypes (237, p. 1 121) 8-8l 4:°'°6

Transit of 18S2.
1888. iIarkness, from the United States photographs 88424-0012

It may be well to add that Todd's result depends upon 2 1 3 photographs, Obreciit's
upon 82 daguerreotypes, and that of the present writer upon 1475 photographs. The
multiplication of the square roots of these numbers by the respective probable errors
of the results gives zbo'496", -^ 0*5 44", and ±0*461" for the probable error of a
single picture.

Discussion of Results. — To facilitate the determination of a definitive value from the
foregoing results, they have been re-arranged in Table IV, the construction of which
will now be explained.

Of the many reductions of the observations of the transits of Venus in 1761 and
1769, all made prior to Excke's time are more or less incomplete, and will therefore
be ignored. In dealing with the remaining reductions of contact observations it must
be borne in mind that within certain limits the value obtained for the parallax depends
upon the meaning attached by the computer to the records made by the various
observers, and as these records will frequently bear more than one interpretation, the
mean of the conclusions reached by several thoroughly competent computers must
generally have a higher degree of probability than the conclusion of any one of them.
Accordingly, the numbers entered in Table IV are, for the transits of 1761 and 1769,
the mean of the results obtained by Encke, Powalky, and E J. Stone; for the transit
of 1874, the mean of the five results obtained by Airy, E. J. Stone, Tupman, and
Puiseux; and for the transit of 1882, the mean of the results obtained by E. J. Stone
and Cruls, giving the former double weight.

54

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Table IV. — Values of the Solar Parallax obtained by Trigonometrical Methods.

Transits op Venus, contact observations.

Transits of 1 76 1 and 1769

Transit of 1 874, English and French observations

Transit of 1882, English and Brazilian observations . . . .

Transits of Venus, photographic observations.

Transit of 1874, United States and French photographs . . .
Transit of 1882, United States photographs ,

Oppositions o/A/ars.
Opposition of 1832 :

Henderson

Taylor

Opposition of 1 849-' 50:

Gilliss and Gould

W. C. Bond

Opposition of 1862 :

Newcomb

Asaph Hall

Opposition of 1877:

Eastman

Gill

Maxwell Hall ,

Oppositions of Asteroids.

Flora, in 1873. Galle ,

Juno, in 1874. Lord Lindsay and Dr. Gill ,

8771
8-859
8-824

8-859
8-842

9-028
9-253

8-495
8-605

8855
8842

8-953

8-78

8-789

8-873
8-765

From the photographs of transits of Venus the results given are, for the transit of
1874 the mean of the values found by Todd and Obrecht, giving the former double
weight, and for the transit of 1882 the result found by the present writer for the U. S.
Transit of Venus Commission. *

The early observations of Mars for parallax have been ignored because they were
made with insufficient instrumental appliances. With respect to the values entered in
Table IV, for the opposition of 1832 there exist only the determinations by Henderson
and Taylor, and for the opposition of 1849-50 only the determinations by Gilliss
and Gould, and by W. C. Bond. For the opposition of 1862 the results obtained by
E. J. Stone and Winnecke rest upon but a small part of the data used by Newcomb,
and therefore only the results obtained by Newcomb and A. Hall require consideration.
Similarly, for the opposition of 1877 we have to deal only with the results obtained by
Eastman, Gill, and M. Hall, because Downing employed a very small part of the
data used by Eastman, and E. J. Stone's paper is virtually an indorsement of East-
man's result.

It is believed that the numbers in Table IV fairly represent all the material now
in existence for the trigonometrical determination of the solar parallax. What is the
most probable result that can be obtained from them % The arithmetical mean of all
the values gives

49 = 8-837" ±0-0614" (142)

ON THE SOLAR PARALLAX VND ITS RELATED CONST AN 55

The means of the results from observations of Mars arc, from the oppositions of
1832 and 1849-50,

1> = 8-845"

and from the oppositions of 1862 and 1877

p = 8-844"

Nevertheless, the four values resulting from the oppositions of 1832 and 1849-50 are
so discordant that they should probably be rejected. Doing so, the arithmetical mean
of all the other values in Table IV gives

o'

p- 8-834" db 0-0086" (143)

Again, taking the means according to the methods of observation, we obtain

//

From transits of Venus, contacts |)=:8 8i8

From transits of Venus, photographs _p= 8*850

From Mars j} — 8844

From Asteroids ^-=8*819

and the arithmetical mean is

p- 8-833" ±0-0056" (144)

Finally, considering only the results which seem most likely to be free from con-
stant errors, we have

//

From photographs of transits of Venus, 1874 . . ^ = 8-859

From photographs of transits of Venus, 1882 . . ^ = 8-842

From Gill's observations of Mars ^ = 8*780

From Galle's observations of Flora p 1=8-873

. From Gill's observations of Juno p = 8-765

and the arithmetical mean is

p = 8-824" "± 0-0146" (i45)

Except in the magnitude of their probable errors, these four means scarcely differ
from each other: but so far as there is any choice among them, well settled principles
would lead to the selection of (143), and accordingly that will be adopted.

56 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

20.— GENERAL FORMS OF THE CONDITIONAL EQUATIONS.

If I is the length of a simple pendulum which makes one vibration per second
of mean solar time, the observed force of gravity will be

g—irH (146)

Upon the assumption that the Earth attracts as if its entire mass were concen-
trated at its center of gravity, its attraction at a point upon its surface in latitude q>
will be

m'/ay (147)

The observed force of gravity is the Earth's attractive force diminished by the
resolved value of its centrifugal force. Putting o for the ratio of the centrifugal force
to the force of gravity at the geographical latitude g>, we have*

4?r2 N cos q> 4 N cos q>

" w ~~~ w

where N cos q> is the radius of the Earth at latitude <p, and tY is the number of mean
solar seconds in a sidereal day. Butf

N = -

V( 1 — e2 sin2 <p)
whence the centrifugal force at latitude q> is

ag _ 4//« cos <p

\t{ s/(\ — e2 sin2 q>)

and the resolved part of that force acting in tlje direction of the vertical is

AQCI COS2 q) /on

ag cos <P — U2 ,/ ~r-^ — r (148)

Equating the Earth's attraction to the force of gravity augmented by the centrif-
ugal force, we have

k2F'

-~ = g(i+a cos <p)

Whence, by (146)

—2— g^(I + <yc°s <P) (J49)

* Compare 14, 2 Teil, p. 84. J 28, p. 323.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

57

If T is the length of the sidereal year, expressed in seconds of mean solar time,
and ax is that value of the semi-major axis of the Earth's orbit which would satisfy
Kepler's third law, then

T!=47'e<) (,5o)

and by eliminating F/7T2 between (149) and (150), and re-arranging the terms

S + E'_ 4«i3 ( n

E' _ lT-a2p\i + ff cos cp) U5U

Owing to the spheroidal form of the Earth, those points which have \/^ for the
sine of their latitude are the only ones upon the Earth's surface at which a pendulum
will vibrate as it would if the mass of the Earth were concentrated at its center of
gravity. In order to justify the assumption made in equation (147), we must there-
fore take sin2 g>=.$, and consequently cos2 <p — §. We also put

a z= r sin p
a, — r (1 +k)

Substituting these values in (148) and (151), they become

S + E'_ 4«(i+tt)3

E' ~lp2T2 sin3 p(i +<**/%)

(152)

G^ = W^¥) (I53)

From (3), (5), (8), (14), and (17) we have

a = 20926 202 feet 1 = 3'256 872 feet

e2 = 0006 803 48 1 019 ^2 =1:7424 25206s8

p2 = 0997 742482

Substituting these values in (152) and (153)

<JV§ = 0-002310461 5

S + E' r -,0+*)3 054)

^— — [7-409 9290] > '. /

E I sin* p

where the quantity in brackets is the logarithm of the number which it represents.

By attributing proper values to the symbols in equation (154) it may be applied
either to the Earth revolving around the Sun or to the Moon revolving around the
Earth. In the former case we shall have from (16) and (25)

T = 3i 558 i49'3H8
(1 + «) = 0-999998 710

and the substitution of these values in ( 1 54) gives

* = [2784 993 2]J(s|eO ('55)

58 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

where $ is expressed in seconds of arc. Of the constants which enter this expres-
sion I is the most uncertain, and as it is not trustworthy beyond the fifth significant
figure, the logarithmic coefficient in (155) is not trustworthy beyond the fifth decimal
place.

The Earth's mass is very small compared with that of the Sun, and if in con-
formity with custom we take the latter for unity, we may write with all needful
accuracy

E' —y

S + E/_

Further, in (155) the symbol E' refers to the mass of the Earth alone, while the
quantity usually called the mass of the Earth is the combined mass of the Earth and
Moon, denoted by the symbol E. Bearing in mind that the mass of the Moon is
expressed in terms of the Earth's mass as unity, it is evident that

F/_E/(i+M)_ E
- x _|_ m - 1 + M

and therefore when E' is changed into E (155) takes the form

^ = [27849932]^— ^j (156)

In order to apply ( 1 54) to the case of the Moon revolving around the Earth we
must change the symbol S into M, (1 + «) into (1 + n'), T into T„ and p into P
Then, from (39) and (57)

T1 = 2 36o59i-5s
( 1 + n') — 1 -ooo 908 743

and by substituting these values we find, after a slight transformation

W_ sin3P

M — [4*665 070 7 — 10] — sin3 P

(157)

where, as in (155), the logarithmic term is not trustworthy beyond the fifth place of
decimals.

I" x 755 D'Alembert determined the Moon's mass from the phenomena of pre-
cision and nutation, but to do this with extreme accuracy seems a difficult matter.
The most recent attempt is by Mr. E. J. Stone,* who states that his equations include
all terms of the third order in the lunar theorv. With some changes of notation, and
after restoring the factor cos &>„, they are

i1 =:(A«+B;ff)cos coQ (158)

$ = C«£ COS OO0

*i87, P- 43-

ON THE SOLAR PARALLAX AND lis RELATED CONSTANTS.

59

where u is a constant depending upon the Sun's mean disturbing force, the momenta
of inertia of the Earth, and the Earth's angular velocity; and

A = i + 3 e,2

2

B=i+3cv-6r

2

C = 2ZfI+3^_5 y>\

fl V 2 2 J

(159)

As Mr. Stone has not published any details respecting these formulae, it may be
well to show how they can be derived. According to Serret*

Azzi+3e,2

2

Bzzi+3e22-3!2

(160)

Czz

P

but that geometer has neglected terms of the third order with respect to the inclina-
tion and eccentricity of the Moon's orbit, and to restore them we must replace I by

(i + ^e22) sin I cos I. Bearing in mind that sin £1 zz ;•', we have

sin I cos I zzsin 2QI) cos 2(.]I)

zz 2y{\ — ;/2)*(i — 2Y2)

— (27 — 473)(i — y2)i — 2y — $y3 — etc.

and therefore to take account of all terms of the third order in the equations ( 1 60), it
suffices to replace I by

(1 + 3 ^(27 — sr5) zz 2;/ + 3Cfy — 5/

(161)

and I2 by 4/2. Upon making the substitution the equations (159) result.

Reverting to the equations (158), eliminating u and e from them, and introducing
the sines of the parallaxes instead of the mean distances, we get

M

AjSshVj?

(C£ - B$) sin3 P
But from (154)

S sin3_p = E/([2,4ii 704 2 — 10] —sin3 p)
which being substituted in (162) gives

(C£ - B%) sin3 P

(162)

e;_

M ""A^([2'4ii 704 2-- 10] -sin:»

(163)

*83, pp. 303, 313, and 315.

60 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

From (24), (61), (64), and (52)

exrrOOi6 771 049
e2z=: 0*054 899 720

y =z 0-044 886 793
M = 0-337815 984

and with these values, from (159)

A rz + 1 "ooo 42 1 902
Br-f 0*992 432 024
0^ + 0-265609855

(164)

In (163) the term sin3jj is so small that we may safely use the value found by
assuming p rr 8-834", and by substituting that, together with the values of A, B, and C
from (164), we shall have

^=sin3P} 10 288 642 1— 38442769 S

(165)

To find the relations existing between |J, ^, and P, we equate the right hand
members of (157) and (165), and thus obtain

1 = 1 J

1 — 216 236 65 sin3 P
3757 444 9 — 807 952-64 sin3 P

(166)

The parallactic inequality of the Moon is given by the expression*

sinQ'^F^^X^sinD
tj + M «!

(167)

where D is the mean angular distance of the Moon from the Sun ; and when sin D
becomes unity, Q' becomes Q. But

«,=<'+*')

fli =

a(i+x)

sin P sin p

Whence, with the numerical values of (1 + h') and (1 + u) from (57) and (25)

«2 ( r + * )sm p n sin p

— 7 s • r> = I OOO Q IO 034 — — £

ay (i+x)sinP y ° sin P

(168)

Delaunay gives f

F v , a2

2423

(169)

*52, T. 2, p. 847, eq. 342; S3, p. 37, and 57, p. 36.

fS2, T. 2, p. 847, eq. 342, and 53, p. 18.

ON THE SOLAR PARALLAX AND [TS RELATED I 0NS1 WIS. 6l

But instead of using the rigorous formula (i68), he arranged his numerical computa-
tions so as to employ the expression

(170)

<h _ y( 1 + >0

a,- P
from which, with

# = 875" P = 3 4227// 1 -f y<! — 1 -ooo 908 743

he obtained

<h/«i = 0*002 558 784
Substituting that value in (169), we find

FrrO'241 086 (l7l)

From the data given by Professor Newcomb,* it appears that the value of F
implicitly contained in Hansen's lunar tables is

F= 122-032" arc i"*£±**X| (172)

but it is not quite clear whether al/a2 should be derived from (168) or (170). If the
former, as according to Newcomb, Hansen used

^ = 8-6085" P = 3 422-25" M=io

(172) gives

Fz= 0-240 922
If the latter

F = 0-240 933 (173)

The mean of (171) and (173) will be adopted, namely

F = 0-241 010 . 074)

Reverting to (167), making sin I) unity, substituting the value of «2/ai from
(168), that of F from (174), and re-arranging the terms, we obtain

V J- M

^ = [5-303 1248- 10] pq|,±^ (175)

*62, pp. 69, 79, and 87.

62 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

where the quantity within the brackets is the logarithm of the number which it repre-
sents, and on account of the uncertainty in F it is not trustworthy beyond the fifth
place of decimals.

The lunar equation of the Earth's motion is*

<yy = -E/+MX^|rXcoss/sin(y/-y) (176)

in which p and P' are the actual values of the solar and lunar parallaxes at the instant
for which Sv is required. For any given lunation, Sv will evidently attain its maxi-
mum value when sin (V — v)—i, that is, when the longitudes of the Sun and Moon
differ by ninety degrees. If now we have an extensive series of observed values of
Sv, covering many complete revolutions of the Moon's node, Sv will have assumed all
possible values, the mean of which will be the constant of the lunar inequality; p'
will have assumed all possible values, the mean of which will be the constant of solar
parallax ; and the Moon will have had all possible latitudes, the mean of which will
be zero. With P' the case is somewhat different. It is equal to the constant of lunar
parallax, plus a series of terms multiplied by factors made up of the mean anomaly
of the Sun, the mean anomaly of the Moon, the mean distance of the Moon from its
ascending node, and the difference of the mean longitudes of the Sun and Moon. All
these terms, except those involving the difference of the mean longitudes, will assume
all possible values and vanish from the mean. The mean of all the values of P' will
therefore be, P -f terms depending upon the difference of mean longitudes! of the Sun
and Moon. Turning now to the lunar theory, we find but a single term of this kind,J
and its value is 28' 178 8" cos 2D. As we have supposed all our observations of Sv
to be made Avhen D was 900, the value of this term will be — 28* 18", and the mean
value of P' will be P — 28-18". However, it will be better to write

P' = P(i + G) (177)

and then, according to Delaunay, we shall have J

PG = - 28-178 8"
which, with P — 3 4227", gives

G=: — 0-008 232 92 O78)

To find the value of G employed in Hansen's tables of the Moon we have,
according to Newcomb§

PG = - 28-225"
which, with P 1=3 422-25", gives

G — — 0-008 247 50 (1 79)

*4r> P- 47- -f Strictly speaking, it should be the difference of true longitudes of the Sun and Moon.

+ 52, T. 2, p. 917, eq. (27). £62, P- IOS-

— 1

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. 63

The substitution of the mean of (178) and (179) in (177) gives

F = 0-991 759 8 P

Substituting the mean values thus found in (176), and re-arranging the terms,
we obtain

* = [4-681 9624- io]PL51+M (i8o)

where the logarithm of the numerical coefficient is trustworthy only to five places of
decimals.

If V is the velocity of light, 9 the light equation, or, in other words, the time
required by light to traverse the mean radius of the Earth's orbit, and a the equa-
torial semidiameter of the Earth, then

whence, with the value of a from (3)

p = [?^A48l6] (l82)

The mean velocity of the Earth in its orbit is*

2 77T

TV(i-0

and if we assume the constant of aberration to be the ratio of that velocity to the
velocity of light, then

2 7tr
tan « = ™-

VTV(i-0
But r zz a/ sin p, whence

^-TVtfar^i'VCi-O
Substituting the values of a, T, and ex from (3), (16), and (24)

[7-5260362] (l84)

Va

*2, vol. 1, p. 637.

/

64 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

21.— THE LEAST SQUARE ADJUSTMENT.

The conditions which must be satisfied by the adjusted quantities are embodied
in equations (156), (157). (166), (175), (180), (182), and (184). For convenience of
reference they are collected in the group (185), where the quantities within brackets
are the logarithms of the numbers which they represent, and although given to six
places of decimals they are trustworthy only to five.

1 sin3 P

M " [4665 070 70 — 10] — sin3 P

i

o or
o or

/ E V

Vi=2>— [2784 993] (7^^)

v2 —p - [5-303 1 25 - 10] PQ — T^j

o or v3=p — [4'68i 962— 10] PL * + (185)

[8-912482]
o or v4=. p — l-

o or v^zzp —

ye
[7526036]

o<** = l-gj 1-216236-65 sin3 P >

I 3757 444 9 — 807 952-64 sin3 P >
If the observed quantities were rigorously exact, their substitution in the con-
ditional equations (185) would reduce all the right hand members of the latter to
zero ; but in general this will not happen, and instead we shall obtain a series of
residuals which may be designated vu v2, v3, etc., as indicated in (185). To make
these residuals disappear, a series of corrections to the observed quantities must be
determined, such that

p=p' + dp P-P' + dP 2» = £' + </? eAc, etc.

where the observed quantities are distinguished by an accent, and the differentials are the
required corrections. The first step will be to differentiate the equations (185), thus :
dM = -3 cotP arc i"(i +M)dP

o = dp + ^? dM - I^549|] dE
1 ^3(i+M) 3/(1 +M)

o = ^-^±I^Mal^dM_[5.303I2]i^(p^ + Q.dP)
o = * +^-[4-68i96]PL,M_[4.68i96]jJfM(p^L + L^p)

o = dP+rdV+2de (l86)

v 0

o-dp +*dV + Pida

+ 3 arc i"([5-334 93] g - [SW 39] I) ^ P C°S P"dP

[0-574 89] - [5'9°7 39 sin3 P]

.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

65

Upon reducing the coefficients in (186) to numbers by means of the data in (195),
eliminating dM, which is not an observed quantity, and adding vu v2, v3, etc., to the
resulting expressions, we obtain

o=vx + dp -[7-411 7
o zr r2 + dp +[8-1170
o=v3 + dp — [9*8134-
o = Vi + dp +[5-6758-
o = v5 + dp +[5-6758-
o = v6 + d§[ — [9*2633

io]^P-[5-98o2](?E
io]rfP — [8-8478— 10] tfQ
10] (/P — [0*144 5] rfL
10] d\T + [8-249 8—10] dd
10] dV + [9-635 1 — 10] da
10] </g + [9-3243— 10] </P

(18/)

From (185) and (195) the reciprocal of M was found to be 83-748, and accord-
ingly that value was used in forming the differential equations (187). The final
solution of these equations gives i/M — 81*197, out an inspection of the equations
(186) shows that the difference between these two values can not sensibly affect anv
of the equations (187) except the third, in which it would have been better to have
used i/M = £(83748 + 81*197) = 82*472.

The equations (187) contain as unknown quantities ten corrections, which must
be so determined as to satisfy these equations rigorously, and at the same time make
the sum of the weighted squares of the corrections a minimum. To accomplish that
we may use either the method of correlatives, or the method of double elimination,
sometimes called the method of independent unknowns. As the probable errors of all
the corrections are required, and the equations lend themselves readily to a double
elimination, that method has been adopted. The equations (187) have been used to
eliminate six of the unknowns, and in (189) each of the ten is expressed in terms of
the remaining four, namely, in terms of dp, d<£, d%\, da, and known quantities: the
coefficients there given being the logarithms of the numbers which they represent.
The weights have been computed from the probable errors in (195) by means of the
formula

Log. weight r= — 4 — 2 (log. probable error.)
Equations ~S~o. (189).

(188)

dp

<*%

d%

da

ft

Log. weight.

dp =

dT —

-\- 00000

0-1310
7-8344—10

I-2IIO
9-9016 IO

6-9iiS — 10
9-5918 — 10
9-9172 — 10
5-9828— 10
2-8874-- 10
c 15888

+ 9-9390— 10
-f-o-oooo

— 0.6757

....

- [06757] n

dq =

</L ==
da —

+ o-oooo

— 99449 — IO
+ 0-3446

-\- 1-1522

+ 9-8555-- 1°

-j- 9-2082 — 10
— 9-6079 — 10

-j- o-oooo

+ 1-3853

— 39593

-|-[l-I522] C; —[9-9449 — I0] 1%

+ [98555 — IQ] *•> + [o-3446] v6

dd —

— [I*7502] (l'4—Vi)

— [43242] f5

-\- [4-0198 — 10] v, + [2-1072 — 10] rfi

dV —

— 4-3242
-(-4-0198 — 10

JE —

-13705 — 10

-+- 2-1072 — 10

698*3

66

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Imagining the symbols dp, dP, d^*, etc., in the first column of the equations (189)
to be replaced by zeros, the weighted normal equations (190) have been formed in the
usual way, but for lack of space on the page their absolute terms are represented by
the letters A, B, C, D, whose values are given in (191).

o = + A + 36-482 dp— o-i 12 63^^ + 0614 29^ + 14-822 dot.
Oz-r + B — 0*11263(^+16*324 (/j? — 0-37844^^ o-ooooofZo:
orr + C + 061429^)— 0-37844(7^+2-8612 dQ O'oooooda:
Or= + D+ 14-822 dp o-ooooo(/g o-oooooc2$+ 7-2804 da.

(190)

A = -fo-425oi vx-\- 0-16451 r2 + 0-20082 v3 + 34'340 vh-\- 0*614 291^

B = — o*ooo 953 02 vx + o-ooi 871 5t;2 — 0*113 55 vs— 0*37844^
C = + 0005 197 6 vl — 0*010207 1*2 + o*6 1 9 30 r3 + 2-0640 v6
D*= — 0*13131 v4+ 14-953 v5

The general solution of (190) is

dp — — 0*161 962 A — 0-00031 2B + 0-034 731 C + 0*329 735 D
d<£ =. — o-ooo 3 1 2 A — o-o6 1 449 B — 0*008 06 1 C + o-ooo 636 D
d£ — + 0*034 731 A — 0008061 B — 0*3580270 — 0*070 709 D
da = + 0*329 735 A + 0000 636 B — 0-070 709C — 0*808 655 D

(190

(192)

By first substituting in (192) the values of A, B, C, and D from (191), and then
substituting in (189) the resulting values of dp, d&, dQ, and da, we obtain the
formulas (193), which are the expressions for the desired corrections to the observed
values of p, P, •£, etc., in terms of vu v2, v3, etc. The coefficients in (193) are the
logarithms of the numbers which they represent.

Formula; Wo. (193).

Vl

Vl

V3

vt

t's

Vt

dp =

— 88367 — 10

— 8-4313—10

— 8-0406 —

10

— 86365 — 10

— 9-8002 — 10

— 84423 -

10

dp =

— 87873-10

— 8-6474 — 10

+ 00065

— 8-6442 — 10

— 98079 — 10

— 01 294

<*£ =

— 6-0641 — 10

— 5-9245 — 10

+ 7-2840 -

■ 10

— 59217 — 10

— 70806 — 10

+ 7-8079-

10

41 =

+ 8-1109 — 10

+ 7-9710—10

— 9-330I -

- 10

+ 7-9678 — 10

+ 9-1315 — 10

— 98540 -

10

dq =

— 99939— 10

+ 1-1400

+ 8-5155-

- 10

— 9-7944— 10

— 0-9581

— 9- 8085 -

10

dh =

— 8-3146 — 10

+ 7-1326 — 10

+ 9-3720 -

■ 10

— 8-0203 — IO

— 91842 — 10

+ 97844-

10

da =

+ 9-1454—10

+ 87401 — 10

+ 8-3494 -

- 10

+ 9-0260 — 10

— 98858 — 10

+ 87510-

10

dd =

+ 05307

+ 0-1254

+ 97347-

- 10

— 17298

+ '•5751

+ 0*363

dV =

+ 22446

+ 1*8393

+ 1-4488

— 17278

— 2S9I9

+ 1-8506

dE =

+ 3-9890—10

— 2-4493 — 10

— 21533-

- 10

— 26551 — 10

— 3SI88 — 10

— 2-4038 —

10

For computing the probable errors of the adjusted values of p, P, ;£, etc., we
shall need a series of formula3 expressing each of these adjusted values in terms of
the originally observed values. As a first step toward finding them we substitute in
(193) the values of vlt v2, v3, etc., from (187), and thus obtain the equations (194).

ON THE SOLAR TARALLAX AND ITS RELATED CONSTANTS.

67

Equations No. (194).

c/5

0
u

-

Logarithmic coefficients for computing —

dp

dv

4t

*%

dq

dp
dP

,/Q
</L

dd

dV
dE

+ 98928 — 10
-7-0515— 10

— 7-7056 — 10
+ 84423 — 10

— 72791 — 10

— S1S51 — 10

+ 94353 — 10
+ 68S63 — 10
+ 5-5051-10

— 4-8169

— 93480 — 10

+ 99755 -Jo

- 9-3927 — 10
+ 0-1294

- 74952 — 10
+ 0-1510

+ 94436 — 10
+ 6-8940 — 10
+ 55125 — 10

— 47675

—6-6395 — io
— 6-0154 — 10
+ 7-0712 — 10

— 7-8079 — 10

— 47723—10
+ 7-4285 — 10
+ 67163—10
+ 4-1715 — 10
+ 27S55 - 10

— 2-0443

+ 8-6715 — 10
+ 8061 1 10
—9-1173— 10

+ 9-8540 — 10
+ 68188— 10
— 94746 — 10
— 87672 — 10
— 6-2176 — 10
— 4-8361 — 10
+ 4091 1

— 04980

— 84178— 10

— 90718 — 10
+ 98085 — 10
+ 99878 — 10
+ 8-66oo — io

+ 05938

+ 80442 — 10

+ 66627 — I0

— 59741

m

O

Logarithmic coefficients for computing —

dL

da

dd

dW

dE

dq

dL

da
dd
JY
dE

— 87237—10
+ 8-3940 — 10
+ 9-0477 — 10

— 9-7844 — 10
+ 5-9804 — 10
+ 95165 — 10
+ 8S199— 10
+ 62701 — 10
+ 4-8SSS — 10

— 4-2948

+ 9-6488 — 10
+ 73612—10
+ 80143 — 10

— S7510 — 10
+ 75879 — 10
+ 8-4939 — 10
+ 95215 — 10

— 7-2758— 10
+ 5-4971 — 10
+ 5-I256

+ 10341

+ 8-7462 — 10
+ 93996 — 10
—01363
+ 89732 — 10
+ 9-8792 — 10
— 12108
+ 9-9796—10
+ 6S822 — 10
+ 6-5109

+ 2-7484
+ 04592

+ I-H39
— 1-8506
+ 0-6871

+ 1-5933
+ 2-5276

+ 9-9776— 10
+ 8-5965 — 10
+ 8-2248

-33587 — 10

— 1-0131 — 10

— 1-6671 — 10
+ 24038 — 10

— 1-2971 — 10

— 22978 — IO

+ 34545 — 10
+ 0-9049 — 10
+ 95234 — 20
+ 9-9692 — 10

dM = + [6948 04—10] dP

The tabular form has been adopted to save space. It is to be read thus

^ = + [9-8928 — 10] dp —[7-0515— 10] dP — [77056— T°] r?^ + [8-4423 — 10] d%
— [7-2791 — 10] dQ — [8-1851 — 10] tfL + [9 4353— 10] dec +[6-8863 — 10] tf0
+ [5*5051- 10] dV- [4-8169] dE

and similarly for the other quantities. The further consideration of these equations is
relegated to page 71.

68 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

22.— NUMERICAL VALUES OF THE CORRECTIONS BY ADJUSTMENT.

The adopted values of the observed quantities arrived at in the preceding pages
are marked numbers (66), (67), (75), (89), (90), (91), (92), (101), (132), and (143).
For convenience of reference they are all collected in the group (195), where those
which vary with the time are reduced to the epoch 1850X).

p = 8-834" ± 0-008 6"
P = 3 423-08" ±0-121"
g= 50-3586" ±0-00248"
<&= 9-233 1" ±o-on 2"

Q= I25-46" ±0-35" ' , ,

L- 6-514" ±0-016" V yb)

a= 20*466" ±o-on"
6= 49708 ± 102s

Vr 186347 ± 36 miles
E = cooo 003 005 09 7 ± o-ooo 000 016056

By substituting these values of the observed quantities in the equations (185),
we obtain the following results :

1/^1 = 837531

«?!=: + 0-072 86" V4 rr + 0-007 20"

v2 — — 0-00532" v5 — + 0-02998" (196)

t>3 = — 0-252 22" #e = + 0-10638"

Whence, by (193)

dE = + 0-000000052 007 E + dE = 0-000003 °57 io4 (T97)

The value of E given in (195) was obtained from the normal equations (130),
and as they must be satisfied with respect to v and v\ the value of k" deducible from
the corrected E of (197) must now be introduced in them, and they must be re-solved.
Thus new values will be found for the masses of Mercury and Venus, which in their
turn will affect the planetary precession, and through it the entire system of corrections
by adjustment. To take account of these changes, some subsidiary formulae are needed,
which will now be investigated.

Reverting to the normal equations (130), if v" is regarded as known their general
solution will be

ioov r= — 3'56i 599433A — cooo 266658B
ioov1 = — o-ooo 216 658 A — 0-013 762 85 2 B

where

A = + 19-8172"— 21-312 0"^°
B = — 227771 8" + 4 178-625 2'V1

We therefore have

v — — 0-705 3 1 5 797 + 0-749 994 745 v*
^ = + 0-031 304960 — 0575051 828^"

(198)

(199)

ON THE SOLAR PARALLAX ANU ITS RELATED CONSTANTS. 69

But

E + dE = E(i + ya) r" = dE/E (200)

and from Table III, E —0*000002 817409 ; whence, by substitution in (199)

v — — 0705 3 1 5 80 + 266 200dE
y1 — + 0-03 1 304 96 — 204 lojdE

(201)

As these equations are of the form

the equations for correcting the masses of Mercury and Venus will be of the form

m + dm — m{\ + « + b.dEi)=z m{i + a)-\- mb.dE (202)

where m is the provisional mass of the planet. With

m — o"ooo 000 333333 m[ = o-ooo 002 488 509

from Table III, and the values of m(i -\-d) from (132), a comparison of (201) and
(202) gives

Mass of Mercury = o-ooo 000 1 14 882 + 0-088 733 dE , .

Mass of Venus =0000002 471 082 — 0-507 92 2(?E

Finally, if instead of the values of v and v1 formerly used, those from (201) are
substituted in (76), we shall obtain

g = 50-358 6#/- 31 7i6"dE (204)

With the value of dE from (197), equation (204) gives

g = 50-358 6" — o-oor 6" = 50-3570" (205)

which must be used instead of the value given in (195). With it we find

v6 — + 0-10666" (206)

A repetition of the computation of dE then gives

dE — + o-ooo 000 05 2 000

and no further change occurs in |£.

With the residuals from (196) and (206) the formulae (193) give

dp rz — 0-024 2 78" dE — — cooo 644"

dP = — 0-423 53" da=i— 0-012018"

rfg = + 0000 155" dd — + 0989 f (207)

eZltzr — 0017 253" dV = — 10-855 miles

dQ = — 0-498 87" dE = + o-ooo 000 05 2 000

7o

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

The addition of these corrections to the observed values in (195) and (205), and
the substitution of the corrected values in the equations (185), lead to a more exact
mass of the Moon and a second series of much diminished residuals, namely

i/M = 8rio74
#! = — cooo 578" #4 = — 0000058"

V2 = + OOOO 02 2" V5 = + 0"000 02 2"

#3 — — o-ooo 528" v6=z — 0000353"

To reduce these residuals still further they were substituted in the formulas
(193), and gave the additional corrections

^ = + 0-000043" rfL = — 0000330"

dP — — 0000 038" da — — o-ooo 1 34"

d<g — — o-ooo 003" dG — + o-ooi 238s (208)

rfll = + 0000 360" dV z=. — 0*153 8 miles

dQ = + o-ooo 920" iEz:- 0*000 000 000 560

When the sums of the respective corrections in (207) and (208) were added to
the observed values in (195) and (205), the corrected values so found sufficiently
satisfied the equations (185), the largest residual being 0-00002", which corresponds
to an error of one unit in the sixth decimal place of the logarithms, while the constant
logarithms which enter the equations are not trustworthy beyond the fifth decimal
place.

It may be well to note that log. M and log. -iX can not be found accurately to six
places of decimals unless P is known to 0000 1", and certain of the logarithms to at
least eight places of decimals. To find log. sin P with the requisite accuracy we have

log. sin (3 422" -\-x) — 8-219 834 9494 + o-ooo 126 90082; — 0-000000018 5a;2

where x is expressed in seconds of arc. The other logarithms necessary to at least
eight places are

log. 216 236-648 = 5-334 929 300

log. 3757 444 92 = 0574 892 623

log. 807952-643 = 5-907385906

to

It is now possible to improve our values of the probable errors of the observed
quantities (195), and for that purpose we remark that the expressions (189) are in
reality a series of ten observation equations, which determine the four unknowns p, jag,
Q and a from the above-mentioned observed quantities. The corrections by adjust-
ment, dp, dP, <V-£, etc., are the residuals of these equations, and the probable error cor-
responding to weight unity can be computed from them by means of the well-known
expression

067449

f / [pvv] \
\\m — n)

where [pvv~\ is the sum of the weighted squares of the residuals, m the number of ob-
servation equations, and n the number of unknowns they contain. Accordingly, with

ON TIIK SOLAR PARALLAX AND ITS RELATED CONSTANTS.

71

the corrections by adjustment from (207) and (208), and the corresponding weights
from (189), we find for the probable error of an observation equation of weight unity
io'016 755. But by equation (188) we assumed that probable error to bedbcroi,
and therefore the probable errors in (195) should all be increased in the ratio of

i'675 5 = I-

The probable errors of the adjusted quantities next require consideration, and to
avoid ambiguity in the symbol dx, which has hitherto been used to denote sometimes
the differential of ./', and sometimes the correction by adjustment to the observed value
of x, we shall now inclose it in parentheses whenever it has the latter signification.
The adjusted value of any observed quantity, x, will then be x-\-(dx), and the first
step towards finding its probable error will be to express d[x -\r (dx)] in terms of all
the observed quantities which it involves. But d[x + (dx)~\ z= dx + d(dx) and the equa-
ions (194) are of the form (dx) = <i(dx) + b(dy) + c(dz) -\- etc. Further, as the (([na-
tions (187) were obtained from (185) by reversing the signs of the v'a and then
differentiating, and as the values of the v's from (187) were substituted in (193) in
order to obtain (194), it follows that the latter equations are identical with the general
expressions for — d(dx). Accordingly

d(dx) =z — a.dx — b.dy — cxlz — etc.

and by substituting that value in the expression above

d[x -f {dx)~\ ■=. ( 1 — a)dx — b.dy — c.dz — etc.

Then putting qrx, qrs, qr„ etc., for the probable errors of x, y, z, etc., the probable
error of an adjusted quantity, x-\- (dx), comes out

2VE0 - «)V + (K-)2 + (c>\Y + etc.] (209)

The coefficients a, b, c, etc., are given in (194); the primitive probable errors
rxi *Vi *'z. etc' a,'e appended to the observed quantities in (195); and q is the ratio
r' /r", r' being the probable error found from the corrections by adjustment for a
quantity of weight unity, and r" the probable error assumed for such a quantity in
equation (188), namely croi. In Table V the columns headed R0 contain the primi-
tive probable errors >\, rv, rz, etc., or in other words, the primitive probable errors of
p, P, g, etc. ; while the columns headed Ra contain for each of these quantities the
numerical value of the coefficient of q in the algorithm (209). Thus the corrected
probable error of the observed value of any one of the quantities p, P, ^, etc., will
be qR0j and the probable error of its adjusted value will be #Ra.

Table V. — Constants required for computing the Probable Errors of the Observed and Adjusted

Quantities.

X

Ro

R.

X

R»

R.

//

//

p

J- 00086

-J- 0-004024

a

4: 0-OII//

4 0-008 987

p

•121

•028 324

6

I-02s

021854

3?

•00248

•002 479

V

36 miles

3S-283

%

Oil 2

005 983

Q

L

•35

J- 0-016

■05S 169
J- 0.013 "3

E

, 16056
^ 10'-'

, 42038

± 10"

72

ON THE SOLAR 1'ARALLA.X AND ITS RELATED CONSTANTS.

The value of q used in computing the probable errors given in the second and
fourth columns of Table VI was that found above, namely, 1*675 5-

From (207) and (208)

dE — + crooo 000 05 1 440

which gives, when substituted in (203)

Mass of Mercury — 0000 000 1 19 446 , .

Mass of Venus zz o*ooo 002 444 954

By comparing the adjusted mass of the Earth in Table VI with Le Vekkier's
value in Table III we find

^" = + 0*084875017 ±o-oo2 499 980 (211)

and the substitution of that value in (199) gives

v = — 0641 659980 . .

vK — — 0-017 502 574

These values of v, v\ and vil satisfy the first two of the normal equations (130),
and when substituted in the weighted observation equations (129) they leave the fol-
lowing residuals :

+ 0-093 i"± 0*023 1" +0-0026" ±0-087 9" +0-1388"

— 1-048 6" ±0 1679" — 0-62 5 8" ±0-020 1" +0-0298"

— 0-5271" ±0-0260" —0-869 4" ±0010 8" —0-7827"

— 0-6104" ±0*003 3" — 00264" ±0-0060" — 0-467 5"
+ 0-245 9" iooooo" +0-1304" ±0*012 8" — 0-3723"

— 0-2241" ±00108" +o-2i65"±oo2i 1" —0-8443"

-0-8465"

Taking into account the probable error of v[\ these residuals give +0-385 005"
for the probable error of any one of the weighted observation equations (129);
whence, with the weights from the general solution (198)

Probable error of v — ± 0-072 659 . .

Probable error of vi — ± o 004 517

The data employed in this section, the corrections resulting from the least square
adjustment, and the corrected values of the quantities investigated, are all brought
together in Table VI. In computing the distances of the Sun and Moon from the
Earth, the value used for the equatorial radius of the latter body is that given in (3),
and the probable errors of the distances have been found from the formula

dD — D cot p. arc 1" dp (214)

where D is the distance of either the Sun or Moon, p the corresponding horizontal
parallax, and dp its probable error.

ON THE SOLAR PARALLAX AND IIS RELATED CONSTANTS 73

TABLE VI. — Results for the Epoch lSoO-Q,upon the assumption that the Earth's Flattening is 1 : L'03-47.

Quantities.

Observed values.

Corrections by ad-
justment.

Adjusted values.

// //

//

// //

P

8834 + 0014 41

— 002424

8-809 76 -J- 000674

P

342308 + 020274

— 042357

3 422656 43 + 004746

i1

50-357 0 -|- 0-004 J6

4- 000015

5o-357i5± 000415

1

9-2331 ± 001877

— 001689

9-2l6 21 + 0-01002

Q

125-46 + 05S642

— 049795

1 24-962 05 + OO97 46

L

6-514 -(- 0-02681

— 0000 97

6-5I303± OO2I97

a

20-466 + 0-01843

— 0-012 15

2045385+ OOI506

e

4970s + I-7090I*

4- 0-99094"

49799094+ O36616

V

1S6 347 + 60-318 miles

— 11 -009 miles

186335-99 + 591 17 miles

E

0-000003005097
J- o-ooo 000 026 902

} +51440 |

0000003056537
+ o-ooo 000 007 043

M

0-012315 7 +0000 042 11

Mass of MercuiT = °-358 340^o-072 659 ^ i

3000000 8371937^1770352

Mass of Venus = ggg2497 ±Q-oo4 5i7 =

401 847

Mass of Earth zz^84 8?5 ± 0"002 5°o_

Mass of Moon zz

354 936
1

409 006 i 1 880

1
327 168 ±754

S1197 3 ±0-277 6

Mean distance from Earth to Sun zz 92 793 500 ± 70 993 miles.
Mean distance from Earth to Moon zz 238 857 ± 3 -3 12 miles.

23.— ADDITIONAL FORMULA FOR PRECESSION.

If we put <p" for the inclination of the moving ecliptic to the fixed ecliptic of
1850-0; 9" for the longitude of the heliocentric ascending node of the moving ecliptic,
counted from the equinox of 1850-0; and t for the time, counted in Julian years from
18500; then, according to Le Yerrier*

p" zz <p" sin Q" - (g + r ) t + r t2
q" zz <p" cos Q" - (g[ + T') t + r't2

(215)

where

g zz + 0-058 88"
//zz-o-475 66"

fz + O OO6 2 7' ' V + 0-075 62" Vi + 0-007 2,2," v™

— 0-024 96" viv — 0005 40" vv + o-ooo 02
r — — 0-005 25" v — 0-288 79" v* — 0-008 32" v

— o- 1 60 09" yiv — 00 1 3 1 3' ' yv — o-ooo 08" v
r =28„p"
r' zz 28J'

" yVi

// ym

" -.^vi

(216)

*4I, pp. 49-50, and 8, T. 2, p. 172.

74 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Le Verrier has given only the numerical values of r and r' which correspond to
his adopted masses of the planets, and we have now to find the terms involving the
variations of these masses. For that purpose we have the equations*

<W' - -"^-mXZ cos t'-x sin r')

2JU

S2q" - + mn\ m-(Z sin t'-x cos r') (21 7)

2JJ. \ U

t, — \(d$ — <S/)sin t' + %{6q — dq')cos r'
X = £(£jJ — <5y )cos r' — ^(Sq — <5gr')sin r'

whence, by eliminating B, and x

mn ,,-, P t.yN^j

4//

<522"zz+^M(^-<y)*2
4/1

(218)

As here written, these equations apply only to the interior planets. To make
them general, we put

C zz -.11 \J= an

2fX 2fX

and then in (218), instead of mn' /^ju', the coefficient JC must be used for planets
within the Earth's orbit, and JC for those outside that orbit. By employing Le
Verrier's numerical values of C, C, M, dp, and S</,f we obtain the following
expressions :

Action of Mercwry.

+ 0115 39" 4- o-ooo 8" v + 005 7 4" v1

1 oooooo<5,y zz (1 + v ) <( + o-oi 1 7'Vii + o-ooi 6"v"' + 0041 i'Vv J> /

+ C002 6'VV o,oooo"//Vi

+ 0-09517" +o-ooio'V + 00564"^

1 oooooo<V/' =(i -- *" ) { + o-oi4o'V" + o-ooi 5"Vm + oo2i 7'VV \ t2

+ 0000 6' Vv o'oooo";/Vi )

Action of Venus.

^ + 12-63647" — 0-019 9'V + 3-624 5'V ~\

1 oooooo£2/'zz(i + ^ ) < + 4-832 2' V! + 0-1485"^-' 4- 3-8396"^ \ f-

'+ 0-207 3" kv-|- 0-003 o'Vvi j

(220)
r+ 2-72626" — 0*252 5'V +0-949 i'V }

1 000000%" zz (1 +v[ ) < + 1-265 6"^" + 0-083 8' V" + o-668 2"VV >*2

( + 00 1 1 3" 7'v + o-ooo 8" vxi

(219)

*8, T. 2, pp. 55 and 103. fS, T. 2, pp. 93-96 and 100-102.

ON THE SOLAR PARALLAX WD ITS RELATED CONSTANTS. 75

Action of Mars.

/ + 0-38362" +0-0032"?' +0*197 2'V

1 000 000 S.,p" = (1 + ^Hi) J + 0-03 7 o" k" + 0-006 o" v"' + o- 1 3 7 7" "iv

( + 0-002 3'Vv + o-ooo i'Vvl

r+ 0-36157" +0-003 8' V +0-0569"^

] 000000%" =(i+viU)<+ 0-03 26' V" + 0-005 3' Vm + 0-249 2'V

(+ 0-013 5'Vv + 0'000 2"v

J> *M

Action of Jupiter.

+ 5870 87" + 0-089 3'V +4'939 i""1
1 000 000 <5.,/' { + 0-003 6"^" + 0-142 3'VH + 2-737 4" yiv

— 2-055 5'VV + 0-014 5'V-

+ 2-64680" +o-io6 5"k + 1-292 5'V
1 000000%" = (i -- "iv) <^ — 0-000 5'V1 + 0-1240"^" — 0-426 8' Viv

+ 1-552 6'VV — 0-009 6' Vvi

Action of Saturn.

+ 0-64359" + 0-004 i'V +0-228 i'V
1 000 000 %/' = ( 1 - - *-v ) <j 0000 o'V" + 0006 6'V" + 0-386 9'Viv

+ o-oio4'Vv + o-oo6 4'VT'

— 0-14562" + 0-004 9'V +0-059 7" k1
1000000%" = (i--vT)<j o-ooo o'Vsi + 0-005 8'ViU- 0-207 i'V-

— 0-004 3" v7 — 0-004 8" v*

Action of Uranus.

(223)

J' -A

+ 0-005 90" + o-ooo i v + 0-004 O V1
i 000 000 tf^" = ( 1 +>-Ti) <! o 000 o'V" + o-ooo i'Vm + o-oo2 i'V"

— 0000 4' Vv 0000 o'VTI

+ 000152" + o-ooo i'V +o-ooi i'V1
1 000 000 %" = ( 1 + vvi) I 0000 o" v" + o-ooo 1 " vUi - 0000 3" k* V t-

(224)

+ o-ooo 3'VV o-ooo o'V

The factors given on pages 42 and 72 for converting Le Verrier's masses into
those we have adopted are

v =-0-641660 ^ = -0133558 ^=+0-002970

„» =-0-017503 ^ = + 0-002339 ^ = + 0-061947 (225)

v" = + 0-084875

and by substituting them in the equations (216), and (219) to (224), the following
numbers result :

76 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

g + r= + 0-05 2 48 1" g1 + r - - 0-466 543"

Action of Mercury + o-ooo 000 041 1" + o-ooo 000 033 9'

Action of Venus + I275^7 + 29176

Action of Mars + 329 9 + 3126

Action of Jupiter + 5 723 o + 25482

Action of Saturn + 6393 — 151 8

Action of Uranus + 61+ 15

Sum .... +0-0000194981 +0-0000056620

With our values of the masses of the planets we therefore have

cp" sin 6" — + 0-05 2 48 1 " t + 0000 o 1 9 498" t2
cp" cos 0" — — 0-466 543" t + o-ooo 005 662" t2

(226)

If we restore the terms for taking- account of changes in the masses, and put

g =1 + 0-052 481" S — + 0-000019 498"^ + r

g' — — o 466 543" 6' — + o-ooo 005 662" t + r'

then the equations (226) may be written

cp" sin 0" —gt+6t cp" cos Q" = g't + 6't

the solution of which is*

tan OJ'^g/g1

arc 1 #2 + <72
and with the values of r and i"" given in (216), we find from these expressions

cp" = 0-469486" (t— 1850) — 0-000003 447" (t— J^so)2
r+ 0-005 92'V +0-295 43" y1 \
+ 1 + o 009 09" Km + 0-156 30" vir > (* — 1 850)
' + o-o 1 2 44" ^v + o-ooo 08" v™)

^rri73°34/54-6" - 8-790 7" (*- 1850) (22?)

— 2480'V — 18 832" v1

— 2 792"^ + 18760"^

— 3 002" vv — 5" Kvi

To prevent the possibility of misapprehension, it may be well to state explicitly
that the j^s in the equations (227) relate to the values of the masses adopted in the
present paper, and not to Le Verrier's values.

*8, T. 2, p. 104.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

77

Reverting to the equations ( 1 58), and bearing in mind that in tlicm the unit of
mass is the Earth without the Moon, we have in the first equation of the group

M EM

S - 1 + M

r* __shr V
r*~Bm*p

whence, by substitution

e zz-

EM sin3 P

(i+M) sin3i>
and from the second and third equations

(228,

AC cos con

(229)

(230)

With the adjusted values of p, P, ^, Q, E, and M from Table VI, and the values
of A, B, and C from (164), these formula} give

£ zz 2180 260

tfzz 1 7*348 662'

(23O

Passing now to the analytical formulae for precession and nutation given by
Serret in his admirable memoir,* after applying the corrections explained in con-
nection with equation (161), putting

A = 1 + 3 e?
2

B = i+^e22-6y
Czz^(i+^22- \y2)

D = --^e2
2 4

E=--^e22-y2
2 4

2 2

and making some transformations, we have, in our own notation

a zz (A* + Bne) cos a?0 — ^

b = - (Ah + Bus) (j C0S 2G}° + ^ nexe\ cos co0
2 sinfi)0 2

b zz «#' cot 2 q?0 + * ^gie'i cos oo0

2

/zz - (Ah + B«£) # cos G}0 = -ag

P zz a — (#-{- i-") cot ooQ

P' zz & — (ag' + r) cot &><, + 99' c°t2 ^o

Q =// + ^'

Q' =f +r' — ag + -g2 cot g?0 — r' — f+ V cot co0

(232)

(233)

*Ss, pp. 313, 314. 315. and 32°-

j$ ON THE SOLAR TARALLAX AND ITS RELATED CONSTANTS.

iff zz at + bt2 + W

CO=GD0+ft2 + a

^-Vt+Y't'+W

GD1 = GO0 + Qt+Q't2 + n
irr , n COS 2 tt>0 . _ V2 .

W — -+- ueij — — ° sin Q, — he L. cos ea0 sin 2 &
sin GD0 fX

(234)
(235)

— u cos co0 sin 2 O — ue cos oo0 sin 2 (£ (236)

m + a wi + «

+ x — ^ — cos &>0 sin An + «£ , , cos go0 sin A„
wi — xsv m — 73 j

y
.G zz — «£ C cos gd0 cos & + * £ — sin a>0 cos 2 &

+ x sin g?0 cos 20 + he— sin go0 cos 2 (£

m -\-a m -\-a

(237)

The second and third equations of (158) are respectively identical with the first
equation of (232), and with the coefficient of cos Q, in (237), that coefficient being the
quantity known as the constant of nutation. e\ is the yearly variation of ev In equa-
tions (236) and (237) © and (£ denote respectively the mean longitudes of the Sun and
Moon, while A0 and A^ denote the mean anomalies of the same bodies. Usually the
symbols © and (£ are employed to denote the true longitudes of the Sun and Moon,
and on account of the smallness of the terms which they affect, no material error will
arise if they are so interpreted in the present case.

As the numerical values of the quantities entering formulae (232) to (237) are
scattered throughout the preceding pages, they are collected here for convenience of
reference :

ej^z + 0'016 771 049 g zz + 0*05 2 481"

e\z=. — 0^000000424 5 g' zz — 0-466 543"
e2=z + 0-054 899 720 r zz + 0-000019 498"

M — — 0-33 7 8 1 5 984 / zz + O'ooo 005 662"

y— +0-044886 793 co0— 23° 27' 3^'36"

305 2563578

m' - **•&& - 83-996 848 52
27-321 661 16 ° yy * °

iSi == the sidereal motion of the solar perigee in 365^ days zz 1 1-3618" + 0*1375"
==11-4993" according to Hansen*; or 61-6995" — 50-2357" zz 1 1-4638"
according to Le VERRiERf. We adopt the mean, namely, + 1 1-4816"
zz cooo 055 664 of radius.

g/j zz the sidereal motion of the lunar perigee in 365^ days, which is according

to HansenJ — -.(<*>— @ — ^1) zz 216 115-2207" — 69 629-3961" — 50-2230"
zz+ 146 435-601 6" zz 0-709 939 830 of radius.

*54. P» '6. f 41, pp. 102 and 51. % 54, pp. 15 and 1 6.

ON THE SOLAR PARALLAX AM) ITS RELATED I ONSTANTS.

79

In (236) and (237) co0 is the true obliquity of the ecliptic at the instant for whirl
W iind fl are required, and we should there take

oo0 — 230 27 31-36' —0-466 54" (t— 1850)— 0000 000 73" (t — 1 850)3
whence

T COS 2ffl, , . ,. n .

Lo^^T^7T-— °'234 4742 +0-000004 365 (t— 1850)
sin cOq

Log. COSQ30 ==9-962 5337+0-000 00O426 (t— 1850)

Log. sin a\ — 9599 9792 —0-000002 264 (£— 1850)

These numbers must now be substituted in the formulas under consideration, and
we shall follow Le Verrier and Serret in first giving the results for luni-solar pre-
cession, and for nutation, with i1, n and e retained as symbols. They are

" = + [9'962 7169— 10] % + [9-9592345— 10] ne — g

l =— [4*290 2802 — IO] H — [4-28461 I7—IO] HE

h — — [4-325 3772- 10] g— [1-99106— 10] n (238)

/— + [3-067 264 — IO] K + [3-063 782 — IO] tf£

/= + [3-104 547— 10] g

1F= — [9-658 7186 + 0-000004365 (7 — 1850) — 10] ne sin£
+ [773809 +0-000000426(7— 1850)— 10] ne sin 2 &

— [8-863010 +0-000000426 (/— 1850) — 10] h sin 2 0

— [7-732 19 +0-000000426 (t— 1850) — 10] ne sin 2 (£
+ [7-86605 +0000000426 (t — 1850) — 10] h sin A0
+ [7-258 65 +0-000000426 (t— 1850) — 10] ne sin Atf

/2 — + [9-386 7779 + 0-000000426 {t— 1850) — 10] ne cos&

— [7'375 54 — 0-000002 264 (t— 1850) — 10] ne cos 2 &
+ [8 500455 - cooo 002 264 (t— 1850) — 10] n cos 2 0
+ [7-369 63 — cooo 002 264 (t— 1850) — 10] ne cos 2 (£

(239)

(240)

After substituting the adjusted value of js? from Table VI, and the values of n and e
from (231), the formula'. (238) to (240), in connection with (233) to (235), give

</,__$ +50-357 1 5" + 0-0144" ^ +0-1743" ^ I (t_lSr0)
( + o-o 1 69" vm — 0-05 75" v* — o-oi 24" vT $
— 0-00010669"^ — i85o)2+ W
co — 230 27' 31-47" +0-00000641" (7— i85o)2 + /}
if>1 = 50-236 22" (t— 1850) +o-ooo 1 10 22" (£—1850)2+ F
»-„<> , ,,_ < +0-46654" +0-0053"- +02888"^)

13 27^47 J+0.oo83'Viii+o-i6oi"^ + o-oi3i/VMv D y

— o-ooo 000 73" (* — 1 850)2 + &

80 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

W-— {17-2382" +o-ooo 173 2" (t — 1850)
+ { 0*2070" + o*ooo 000 2" (t — 1 850)
— { 1-2655" + 0*000001 2" (t — 1850)
— { 0-2042" + o*ooo 000 2" (t — 1850)
+ { 0-1274" + o*ooo 000 1 " (t — 1850)
+ { 0-0686" + o-ooo 000 1 ' (t — 1850)

} sin

9,

^ sin

2&

} sin

20

} sin

2d

} sin

A0

} sin

A<

(242)

fl = -\-\g-2\6 2" + 0000 0090" (t — 1850) I COS&

— {0*0898" — O-OOO OOO 5" (t — 1850)} COS 2& , .

+ {0-5492" — O-OOO 002 q" it— 1850)} COS 2© ^ ^

+ {o*o88 6" — o-ooo 000 5" (t— 1850)} cos 2t£

Respecting co0, it is to be remarked that Le Verrier's data* give g?i****z 23 ° 27'
49*804" for the epoch 1810*7, ana" by bringing that up to 1850*0 with our yearly
motion, — 0*466 543", the value of go0 given in (241) is obtained. Of course the vs
in (241) relate to the masses of the planets adopted in this paper, and not to
Le Verrier's masses.

It yet remains to deduce from the group of formulae (241) the ten quantities
usually employed in computations relating to precession, and these quantities will be
given in a shape permitting of their ready reference to any desired equinox and
ecliptic. The transformation from one equinox and ecliptic to another might be
effected by Hansen's formulae,! but they involve the constants g, [/', r and /, whose
values are not always known, and to avoid that difficulty we shall develop formulae
involving only the coefficients in equations (234) and (235).

If dip /dt is the annual change of rp at the instant tQ, then by (232) and (234)

dip cos G?,

+— =a

dt cos G?0

rP =a^^}(t-t0) + b(t-toy (244)

COS 6O0 V 7 V 7

But by (235), which relates to the epoch T,

©, = ®o + QCo - T) + Q'(to - T)2 (245)

whence, with sufficient accuracy

cos o?! — cos go0 — Q sin Go0(t0 — T)

and by substituting that value in (244), the expression for ip given in (273) results.
Further, as in the second equation of (234) a>0 is the true obliquity of the ecliptic at
the instant T, in order to change the epoch from T to t0 we have only to substitute ^
for go0; and by so doing the expression for a> given in (273) was obtained.

*4"> P- 51- t 78, cols. 114, 139, and 154.

<>N rHE SOLAR PARALLAX AND ITS RELATED CONSTANTS. 8l

The expressions for ^x and cou in (235), are of the form

u =A + B(*-T)+C(*-T)2

where

dT '-*W2

and consequently, the values which they assume when the origin of time is changed
from T to t0 may be found by the algorithm

u' ■- A + B(*0 - T) + C(t0 - T)2

+ [B + 2 C(t0 - T)] (t - t0) + C(* - t0f

The results are given in (274).

If A is the planetary precession during the interval (t — t0), ami ^(a^-f-co)
— co0 -f- daoQ, then *

\—(if>— fa) sec Oo + da>0) ^^

— (if>— ipi) sec w0-\-(ip — Jp^dcoo sec go0 tan co0

But from (273) and (274)

xP-tPi = [a- P - (2P + «Q tan ®0)fo-T)] (*-*,) + (&- P')('-*o)2
i(«h + ®) = *>o + Q(f. - T) + QU - T)2 (247)

+ [£Q + QU-T)](*-*o) + K/+Q0(*--*o)2

whence

^0zzQ(^0-T) + Q'(^o-T)2+[^Q+Q/(^o-T)](^-g + K/+Q')(^-^o)2 (248)

and by substituting these values of O — ^1) andda>0in (246), and rejecting all terms
above the second order with respect to (t0 — T) and (t — t0), the expression for A given
in (273} results.

To find m and n we have the well-known formulae

But from (273)

m _ *+ cos 00 - d* n= ** sin co (249)

dt dt at

dtp

-a-aQ tan co0(t0 - T) + ib{t - Q

(If

- + sec co0(a - P) - sec a>0(2P' + PQ tan oo0)(t0 - T)
dt

+ 2 sec o?0 \b - P + JQ(a - P) tan «?„](< -f0)

go = a>0 + Q(*0 - T) + Q'(/u - T)a +./(/ - to?
sin a> - sin g>0 + [Q(*0 - T) + Q'(* - T)2 +./(/ - *0)2] cos co0
coa o - cos «0 - [Q(*0 - T) + Q'(/0 - T)2 +f(t - t0f] sin oo0

*2, VOl. I, p. 607.

6987 6

g2 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

and by substituting' these values in the formulas (249), and rejecting all terms above
the second order with respect to (t0 — T) and (t — t0), we find

m = + a cos co0 — (« — P) sec ft>0

+ [(2P' + PQ tan co0) sec oo0 — 2aQ sin Go0](t0 — T)
-)- a sin &>0(Q2 tan co0 — Q')(t0 — T)2 . .

_ ( + [2{b - F) + Q(« - P) tan ©0] sec <*>0 ) _
( — 2&cosft>0 + 2&Q sin &>0(70 — T) )

— af sin cQ0(t — t0f

n z= + a sin ft>0 + «Q(cos ft>0 — sin go0 tan Go0)(t0 — T)

— rtcosft)0(Q2 tantt»0 — Q')(^0 — T)2 , .
+ 2?;[sin ft>„ + Q cos co0(t0 — T)](* — 10)

+ af cos GQ0(t — 10)2

When converted into numbers by means of the coefficients in (241 ), these expres-
sions become

m = + 46-063 1 5"

+ o-ooo 277 23" (t0 — 1850) + 0-000000000 1 15" (t0 — 1850)2

+ [o-ooo 277 29" —0000000000 192" (t0— i85o)](^ — ^0) (252)

— o-ooo 000 000 623" (t — 10)2

n — + 20-04661"

— 0-00008481" (t0 — 1850) — 0-000000000 266" (/0— 1850)2

— [0-00008494" —0-000000000443" (t0— iSso)](t—t0) (253)
+ o-ooo 000 001 435" (t — t0)2

It is, therefore, evident that when neither (t0 — T) nor (t — 10) exceeds a century, all
second order terms may be neglected, and the algebraic expressions for m and u may
be written as in (273).

Expressions for q>" and 0" have already been found from g, gf, r and r', but we
have now to deduce them from the coefficients in (234) and (235). For that purpose
we shall employ the equations*

q>"2 = (<»! — go)2 + A2 sin2 h(G>i + 00) (254)

tan (G" + y + hA) = — ^— sin £Oi + <*>) (255)

&>! GO

The first of these equations may be written

>i

(&>! — ft>)

" 1 \S 1 A2 sin2 i(G), 4- go) )\
q> — — (goy — go) X 1+ 2\ i-r M

( (go, — GOV S

whence, expanding by the binomial theorem

" / \ A2 sin2 h(GO, + go) , , ,x

<?>= — (©! — ©)— 2V i-r ; + etc_ (256)

2(&»! — ft))
*2, vol. I, p. 607.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. 83

If now we put

£(<»!+ go) = w0 + dwn

A-a-V _(2P: + PQ tana^-T) (257)

B=6-P' + £Q(a-P) tana>0

we shall have from (248), (273), and (274)

dte0 = Q(*0-T) + £Q(*-*0) + etc.
A = see o?0 A(< — t0) + see &>0 B(/ — /,, ) ^ 5

o»1-G,=zQ(t-t0) + 2Q'(t0-T)(t-t0)-(f-Q')(t-t0y (259)

Whence, with sufficient accuracy

sin ^(&>j -f- &>) — sin oo0 -\- dco0 cos go0
A2 siir i(^ + co) - + tan2 g>0[A2(* + *0)3 + 2AB(* - *0)3] (260)

-|- 2 A2 dco0 tan c»0(£ — /0)2

The substitution of (259) and (260) in (256) gives, after rejecting all terms above
the second order with respect to (t0 — T) and {t — t0)

= - [Q + tan2 oo0{a - P)2/ 2Q](< - tQ)

_ (+2Q'+tana70(a-P)2 I (t -T)(t-t^ (26^

^-tan2^0(«-P)(2P' + PQtana?0)/Qr'° L){* h) (26l)

+ \ (/- Q') - tan2 co0(a - P)|6 - P' + iQ(a - P) tan a>„]/Q j(* - t0)2

and as the quantities involved are so related that the coefficient of (/„ — T) is known to
be exactly twice that of (t — t0)2, (261) naturally takes the form given in (274).
Reverting now to (255); after eliminating A by means of the relation*

if> — ip! = A COS £(60! -f go)
and putting

tan {&' + \4> + Jft) — tan (0O" + dd0") = tan 00" + d90" sec2 0O" (262)

we have

tan 60" +. d90" sec2 6>0" = -^^ tan £(>, + a>) (263)

But, from (247) and (259)

0-^x__ «_P_(2P' + «Q tan Q(70 - T) + (b - P')(t - t0)

G>! — GO

Q\I+f(t0-T)-l^(t-t0)\

"Q" v ° 7 Q

Whence, with sufficient accuracy, through an expansion by the binomial theorem

ip — ipx a—V ( 2P' + aQ tan®0 2Q'(« — P) ? ,', T>.

co1— co Q ( Q ^ ) (26,n

*2, VOl. I, p. 607.

84 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Also, by putting §(col -f go) — go0 -f cIgo0, and taking the value of doo0 from (258)

(265)

tan £(«»! + ct») zz tan &>„ -f- <to0 sec2 go0

- tan g?0 + sec- co0[Q (t0 - T) + §Q(* - f0)]

A comparison of (263) with the result obtained by multiplying together (264)
and (265) shows that, to terms of the first order with respect to (t0 — T) and (t — t0)

tan 90" — tan m0(a — P)/Q

(266)

d90" sec2 0„" = - tan ©0[(2P' + «Q tan *>0)/Q + 2Q'(a - P)/Q2](V0 - T)

+ («-P)sec-'^f/o-T)

+ tano?0[(&-F)/Q + («-P)(/-Q/)/Q2](^-^) ^

+ .K«-P)sec2o?0(^-/0)

As (262) gives

e" = <9o" + d0,"-i(tf + ft)

by taking the values of #0" and d90" from (266) and (267), and remembering that to
terms of the first order

i(* + *i) = *(«+P)(<-'o)

we find

9" - e0"

tan tt>0 cos2 0O"

[2F + «Q tan co0 + 2Q'(a - P)/Q](*0 -T)

+ (a — P) sec2 a>0 cos2 £0"(£0 — T)

-f ta" ^Q08' 9°" \P - P' + (a - P)(/~ Q')/Q](' ~ U)

+ J[(« - P) se°2 «o cos2 <90" - (a + P)](* - /0)

If we had not transformed (255), but had developed

(268)

instead of

G0l — GO

G), — GO

sin £(g>! + o>)

tan £(©! + g?)

we would have obtained the slightly more complicated expression
tan ooc cos2 9n"

9" - 9'' -

Q

[2P' + PQ tan coQ + 2Q'(a - P)/Q]& - T)

+ cos20o,/(«-P)(^o-T)

+ ^ ^q S' *'" [& - P + *Q(« - P) tan «>, + (a - P)(/-Q')/Q](* -'«>
+ J[(a - P) cos2 6?0" - (a + P)](*- f0)

(269)

ON THE SOLAR PARAL1 W AND ITS RELAT] D I ONSTANTS. 85

If M is the longitude of the ascending- node of the mean ecliptic at the time
t + dt, reckoned from the equinox of date t, upon the ecliptic of date t, then

M = fl" + ^* + *

But

e" = e0" + *?ft (270)

and therefore, with sufficient accuracy

M = e0" + 2((°<>r t + ip, (271)

As the relation of M to 0" is the same as that of gd1 to go, when the date of the
equinox and ecliptic is changed from T to t0 in (270) and (271), we must evidently
have

„ (272)

M =6?0"+|2^-, + ^|[(i0-T) + (<--«0)]

and these are the forms adopted in (274); the values of 00" and d90" /dt being taken
from (266) and (268).

rp =4i-Qtan<»0(;0-T)](*-g+^-f0)2+¥f

co = »0 + Q(*0 - T) + QU - T)2 + f(t - t0f + «

A - + sec <*>0[a - P - (2F + PQ tan g?0)<70 - T)](* - 10)

+ sec gj0[6 _ P + iQ(a _ P) tan t»0](< - *0)2 (273)

m = + a cos &>0 — (a — P) sec go0

+ [(2F + PQ tan a>0) secco0- 2«Q sin g>0< \(t0-T) + (f-f0)}
« rr a sin g>0 + 26 sin oo0[(t0 — T) + (t — £„)]

^ =[P + 2P'(/o-T)](/-g + P'(^-g2 + ^

G,1=ffi>0+Q(*0-T) + QU-T)2

+ [_Q + 2Q'{t<)-T)](t-t0) + Q'(t-t0Y + n

cp" = - [Q + tan2 ©0(a - P)2/ 2Q](* - *0)

+ 1 (/- Q') - tan2 G,0(a - P)[6 - F + iQ(« - P) tan g>0]/Q } \ 2&-T)

(t-te) + (t-t0y} {m)

00"= arc tan [tan oo0(a— P)/Q]

A =+t^l^oCOS2^o//[&_F + (r<_p)(/_Q')/Q]

+ i(« _ P) sec2 oo0 cos2 60" - £(« + P)
6" = 00" + A [2(t0 -T)-(t- 10)] + P(*0 - T)
M = 60" + (2 A + P)[a - T) + (* - Q]

86 ON THE SOLAR PARALLAX AND ITS RELATED I I (NSTANTS.

By means of the groups of formula' (273) and (274) the entire system of quan-
tities used in computing precession can be readily found for any desired equinox and
ecliptic when the expressions for tf>, rfv go, and a^ are known for a given equinox and
ecliptic. With respect to the notation, it may be well to remark that ip, od, gou <p", and
0" are the quantities which Dr. Peters designated tf>', 0', 0U tc", and 77. Further, T is
the epoch for which the constants in the formulae (234) and (235) were originally
computed, while t0 is the date of the new equinox and ecliptic to which the precession
is to be referred, and t is the date for which the various quantities are required. In
(273) and (274) all the angles and circular functions are expressed in parts of radius,
and to convert them into seconds of arc it yet remains to introduce the factor arc 1"
wherever necessary.

By substituting the coefficients from (241) in the formulae (273) and (274), all
the following numerical expressions were obtained, except those for <p", 0" , and M,
which are from (227).

ip — [50-357 1 5" + o-ooo 049 43" (/„ — 1850)] (£ — tQ)

- o-ooo 106 69" (t — tQy + V
g? = 23° 2 7' 3 1 -47" — 0-466 54" (70— 1850)

— coooooo 73" (70— i85o)2 + o-ooo 006 41" {t— t0)2-+-fL , .

A =[o-i3i83"-o-oooi8655"(*0-i85o)](7-g {2?5)

— o-ooo 236 52'' (t — t0y

m =346-063 15" +0-000277 23" [(70— 1850) + (7 — /„)]
n = 20-046 61"— o-ooo 084 94" [(£„ — 1 8 50) -f- {t — toy]

^ = [50-236 22"+ o-ooo 220 44" (70 — i85o)](7 — 10)

+ o-ooo 1 10 22" (t — t0f + V
gox — 230 27' 31-47" — O466 54"(70— 1850) —0000 000 73" (t0— 1850)2

— [0-466 54" + o-ooo 001 46" (/(,— I85o)](/ — t0)

-- o-ooo 000 jt," (t — 10)2 + il (276)

cp" = [0-469 49" — O-OOO OO6 89" (t0 — I 850)] (t — t0)

—o-ooo 003 45" (t —t0y
ef'~ 1730 34' 55" + 32-655" (to- 1850) - 8-79i" (t-to)

M = 1730 34' 55" + 32-655" [(t0- 1850) + (*-*„)]

In Section 25 we shall see that the outcome of the present investigation respecting
precession depends somewhat upon the value adopted for the flattening of the Earth,
but formulae (227), (241), (275), and (276) are very nearly correct, and it is desirable
that ready means should be provided for comparing them with the results obtained by
other astronomers. To that end the coefficients of the formulae of La Place,* BESSEL,t
Struve and Peters. J Hansen,§ Le Verrier,|| and the present writer, are collected in

*7, T. 3, liv. 6, chap. 16, converted into sexagessimal seconds by BESSEL, i, p. 2S5, and 1 '<, p. iv.

fi^,p. v.

J109, pp. 190 and 195.

I 78, cols. 113, 114, 139, and 152-154.

|| 8, T. 2, p. 174, and T. 4, p. 104.

ON THE SOLAR PARAL1 \\ AND ITS RELATED CONSTANTS.

87

Table VII, where they are all referred t<> the equinox and ecliptic of 1800; such of the
formulae as were referred by their authors to equinoxes and ecliptics other than those
of 1800 having been reduced to that epoch by means of the formula- (273) and (274),
without the introduction of any extraneous factors. Respecting Hansen's formulas, it
should be remarked that the values of «o„ and P given in Table VII differ slightly from
those in his Tables du Soleil, p. 5; but the P agrees with that in his Tables de la Lune,
p. 16. The dates of publication are, for the article in the Astronomische Nachrichten,
September, 1852; for the solar tables, 1853; and for the lunar tables, 1857.

TABLE VII. — Values given by various Authors for the Coefficients in Vonm/la' (231) and (235); said
Values being all re/erred to the Equinox and Ecliptic of L800.

Author.

La Place.

Bessel.

Struve and
Peters.

Hansen.

I.F. Verrier.

FormuLf
(241).

Date.

1802.

1826.

1841.

1852.

1856.

1889.

a
1 000 J>

+ 50-290 34"
— 0121 79

+ 50-378 26"

0-I2I 79

+ 50-379 8 "
— 0108 4

+ 50-355 93"
— 010674

+ 50368 88"
— 0108 81

+ 50354 68"
0106 69

1 000 /

23°27/5i-95//
+ 0009 84

23° 27' 5381"
-\- 0-009 84

230 27' 54-22"
+ 0-007 35

230 27' 54-80"
-f 0-007 05

23° 27' 5561"
-f 0-007 '9

23° 27' 5480"
-j- 0-006 41

P
1 000 P'

+ 501 1 1 36"
+ 0122 15

— 50223 50"
-(- OI22 15

-f 50241 1 "
+ 0-1134

-j- 50-222 95"
-j- 01 12 07

+ 50-224 43"
-f- 0-11289

-+- 50225 20"
+ 011022

Q

I 600 Q'

— 0521 41"

— 0-002 72

- O48395"

— 0-002 72

- 0473 8 "

— o-ooi 4

— 0467 70"

— o-ooi 40

— 047551"

— o-ooi 49

— 0466 47"

— o-ooo 73

As Dr. Peters's formulae for nutation have been more used than any others, it will
suffice to compare our own with them. Accordingly, formulae (242) and (243) have
been reduced to the epoch 1800, and are given below, side by side with Dr. Petkks's
more elaborate expressions* When extreme accuracy is desired, some of his small
terms may advantageously be added to our formulae.

Formulas (242)
W— — 17-229 5" sin&
+ 02070 sin 2&
— 0204 2 sin 2(£
+ o-o68 6 sinAa

Dr. Peters.

— 1 7-240 5" sin Q,
+ 0207 3 sin 2&

— o-204 1 sin 2(£

+ 0067 7 sin(d — r')

— 0-0339 sin(2([ — &)
+ 00125 sin(20 — Q,)

— 00261 sin (3d — O
+ 001 r 5 sin((X + r')

+ 0-0150 sin(d— 2© + r')

+ 0-0058 sin(£ + £-0

+ 0-005 7 sin (cl — Q, — rr)

4- 0-002 o sin (C — Q> + T")

(277)

♦From 109, pp. 170-172.

88

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

— 1-265 6 sin 2©
+ 0-1274 sin A0

Formulae (243)
Q. = + 9-215 8" cos&

— 0-0898 COS 2^
+ 00886 COS 2(£

+ 0-549 3" COS 2©

+ 0-004 4 sin (2/1' — Q)
+ o-oo6 1 sin(2(£— 2©)

—

0-005 2

sin(3([-20 + r') (277)

+

0-005 3

sin (2©— iT) cout',i-

+

0*002 6

cosT'

+

0*002 O

sin 2.T'

+

0-002 5

sin (£+2© — r')

+

0-002 8

sin(2([-2r/)

+

0-002 4

sin (2d — 2&)

—

0-002 4

sin (2©— 2&)

—

0-002 8

sin (4C - 2J")

—

0-003 3

sin (4C - 2©)

—

1 269 4

sin 2©

+

0-1279

sin'© — T)

—

0'02I 3

sin (© + n

—

0-005 8

sin(3©-r)

—

0*000 5

sin (2© — 2.T)

Dr. Peters.

+ 9 223 1" cosQ,

— 00897 cos 2Q>
+ o'o88 6 cos 2([

+ 0-OI8 I COS(2([ — &)

— 0-0067 cos (2© — &)
+ 0-0113 cos (3d — T')

— 0-005 ° cos (C + r1)

— 0-003 1 COS (([ + £ — f")
+ 0-0030 cos((£ — £ — T')

— o-oo 10 cos (([ — G + r')

— 0-0024 cos(2r' — a)

+ 0-0023 cos (3C— 2© + /"")

+ OC02 3 sili 7 '

— o-ooo8 cos 2V

— O-OO II COS (([ + 2© — r')

+ 0'OOI 2 cos (4d— 2/"')

+ 0-0014 cos(4t— 2©)

+ 0-551 O COS 2©

+ 0-009 3 cos (o + r)

+ 0-0027 cos (3© — F)

(278)

The differential formula? given above will suffice for computing- the precession in
all ordinary cases; but if the general formulae should be required, their computation
may be greatly facilitated by using the convenient expressions given by Professor
Stockwell for the constants which determine the secular inequalities of the nodes and
inclinations of the orbits of the eight principal planets of the solar system.*

*S4, pp. l6i-l64and 171-176.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTAN1 3 89

24.— THE DENSITY, FLATTEN! Nil, AND MOMENTS OF 1NEUTIA OF THE EARTH.

In dealing with tlie equations (291), and in discussing the law of density of the
interior of the Earth, we shall need the best attainable values of the Earth's mean
density, its surface density, and its precessional moment of inertia.

The following is believed to be a tolerably complete list of the more important
determinations of the Earth's mean density hitherto published:

1778. Maskelyne and Hutton, from measurements of the attraction of the plumb line by the mountain Sche-

hallien, in Scotland. (290, p. 783.) 4 %

1798. Cavendish, from the attraction of two leaden balls, each weighing 3484 pounds, the attraction being

measured by a torsion balance. (2S0, p. 522.) 5-48 ^ °'39

1811. Playfair, from his own determination of the mass of Schehallien, combined with MASKELYNE and

HuTTON's determination of its attractive force. (302, p. 376.) 4'7'3

1812. Hutton, from Maskelyne's measurements of the attractive force of Schehallien, combined with Pi.ay-

FAlR's data for its density. (291, p. 64.) 5-

1823. Carlini, from pendulum observations made on Mt. Cenis, and at Bordeaux at the level of the sea.

P- 4o) 4'39

1837. Reich, from the attraction of :\ leiden ball, measured by a torsion balance. (271^.98.) 5-438 J- 0-023

1840. Giulio, from Carlini's pendulum observations on Mt. Cenis, after correcting both Carlini's theory and

his adopted length of the pendulum at Bordeaux. (287, p. 384.) 4-95

1842. SAIGEY, from the pendulum observations made by Bougl'ER ami La CoND AMINE in Peru, on Chimborazo

and at the level of the sea, in 1 737-1 740. (272, parte 2nda, p. 125.) 4-62

1842. Baily, from the attraction of two spheres of lead, each weighing 380-5 pounds, the attraction being meas-
ured by a torsion balance. (271, p. ccxlvij.) 5-675^0-004

1851. Reich, from a rediscussion of the experiments which he made in 1837. (310^.389.) 5-484 _[- 0-020

1851. Reich, from the attraction of a leaden ball, measured by a torsion balance during the years 1847-1849 and

1850. (310, p. 418.) S-583±o-oi5

1856. James and Clarke, from the attraction of the plumb line at Arthur's Seat, Scotland. (282, p. 606.) . . 5316 -J- 0054
1856. Airy, from his pendulum experiments in the Harton Collier)-, England. (270, pp. 342 and 355.) . . . 6-566 J- 0-018

1856. HAUGHTON, from Airy's pendulum experiments in the Harton Colliery. (288, p. 51.) 5'4&o

1865. Pechmann, from deviations of the plumb line in the Alps, 4-7, 5-32, and 613; the mean of which is

(14, T. 2, p. 380.) 5-38

1878. Cornu and Baille, from measurements, with a torsion balance, of the attraction of a mass of mercury

weighing about 26 pounds. (284, p. 699.) 5'5^

1878. Cornu and Baille, from BAILY's experiments, after correcting a systematic error. (284, p. 702.) . . . 5559
1878. Poynting, by weighing with a delicate balance the attraction of a sphere of lead having a mass of 340

pounds. (305, p. 18.) 5 r'9 =t°i5

1881. Mendenhai.i., from pendulum observations at Tokio, and upon the summit of Fujiyama, in Japan.

(299, p. 1 24, and 300, p. 103.) 577

1883. Sterneck, from pendulum experiments made in the mines at Pribram, Bohemia. (314, p. 91. Mr.

Sterneck has made similar experiments in other mines, with very anomalous results. For a review

of them, see 308, pp. 234-237.) 5'77

1883. Von JOLLY, from the change in the relative weight of two spherical bottles of mercury, each having a

mass of il-o pounds, when they were compared, 1st, with both bottles close to the scale pans, and, 2d,

with one bottle close to its pan, and the other suspended 68-9 feet lower down. (293, p. 22.) . . . 5-692 -[- 0-068
1889. WlLSING, from the attraction exerted upon a pendulum by two cylindrical masses of iron, each weighing

715 pounds. (322, p. 141.) 5-579 -(-0012

In several instances two or more of the above results rest upon a single set of
experiments, the reductions having been made by different methods, and in such cases
we have only to consider the one which seems most trustworthy. Alter the applica-

9o

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

tion of that precept, the various results may be classified according to the methods of
observation, as follows:

i. — Results from the attractions of mountains, measured by the deviations of the
plumb line.

1811. Maskelyne, Hutton, and Playfair . . . . 4713

1856. James and Clarke 5*3*6 (279)

1865. Pechmann 5-38

Arithmetical mean 5'T36

2. — Results from the attractions of known masses of metal, measured either by

the torsion balance or by the pendulum.

•

1798. Cavendish 5-48

1 85 1. Reich, from his experiments in 1837 .... 54.84

185 1. Reich, from his experiments made in 1847-50 5583 ( % \

1878. Cornu and Baille 5-56

1878. Baily, corrected by Cornu and Baille ■ ■ . 5559

1889. Wilsing 5-579

Arithmetical mean 5-541

3. — Results from comparisons of pendulum observations made at different dis-
tances from the center of the Earth; namely, at the sea-level and upon mountains, or
at the surface of the Earth and down in mines.

1840. Carlini, corrected by Giulio 495

1842. Bouguer, La Condamine, and Saigey . . . 4-62

1856. Airy 6566 (281)

1 88 1. Mendenhall 577

1883. Sterneck 577

Arithmetical mean 5*535

4. — Results from weighings made with delicate balances of the usual form.

1878. Poynting 5-69 , g s

1883. Von Jolly 5*692

Arithmetical mean 5*691

The arithmetical mean of the 16 values of the density of the Earth given in (279),
(280), (281), and (282) is 54.82, but that attributes too much weight to the discordant
values in (279) and (281). On account of our utter ignorance respecting the internal
constitution of the Earth, it will be safer to base our conclusions solely upon experi-
ments of the second and fourth kinds; and if we take the arithmetical mean of the
eight values in (280) and (282), we shall have

Mean density of the Earth rr 5*578 ± 0*019

ON fHE SOLAR 1'ARAl I \\ AND ITS RELATED <

91

The last four values of (280) are unquestionably the must trustworthy of the
whole series, and the arithmetical mean of them alone givea

>-■■

Mean density of the Earth = 55 70 ± 0-004

Probably it will be best to take the mean of the eight values in (280) and (282),
giving- half weight to the first two in (280) and to those in (282). In that way \v<; find

Mean density of the Earth = 5576 ±0016 (283)

which will be adopted. It is scarcely necessary to add that the unit of density here
employed is that of distilled water at a temperature of 3920 Fahrenheit.

As a basis for estimating the surface density of the Earth, we have the following
numbers:

1811. Playfair's data give for the mean density of Schehallien, 9 933 X4713-T- 1 7 804 (302, pp. 374 and 376) 2-63

1823. Carlini. from the lithological constitution of Mt. Cenis, estimated its mean density to be (279, p. 39) . . 2-66

1852. Plana found, from Carlint's pendulum experiments, for the mean density of the rocks upon which the

plateau of Mt. Cenis rests (301, p. 187) 2-71

1856. Jamfs and Clarke found for the mean density of Arthur's Seat, Scotland (282, p. 603) 2-75

1856. Airy's data give for the crust in the neighborhood of Harton Collier)-, England (270, p. 342) 2-53

1882. STERNECK found for the mean density of the crust over the Pribram mines, Rohemia (313, p. 118) . . . 275

1889. F. W. Clarke found for the mean density of the outer ten .miles of the Earth's crust 2-40

The thanks of the present writer are due to Professor Clarke for his estimate,
which was kindly communicated in the appended letter :

Washington, D. C, November u, 1889.
Dear Prof. Harkness :

In rav estimates relative to the abundance of the chemical elements, I have
assumed, for definiteness, a layer of the Earth's crust ten miles thick below sea-
level. The volume of this, including the ocean and the continents above sea-level, is
1935000000 cubic miles, of which 1633000000 is solid, and 302000000 sea.
Hence, by volume, in round numbers we have 85 per cent, rock and 15 per cent,
water. The density of the ocean is a trifle under 103. The figure 103, then, maybe
taken as near enough for practical purposes.

The density of the solid crust is less easy to determine. The greater part of that
crust is probably made up of plutonic rocks; and the average specific gravity of about
200 of these, representing a wide range of localities and varieties, is 2 716. In the
crust are both heavier and lighter inclusions, and at its surface we have bodies of
rather less heavy sedimentary rocks, which range down to a specific gravity of 2-5, or
lower. Probably 2-60 or 2-65 would be near enough for the whole solid mass.

Now, taking 85 per cent, solid and 15 per cent, liquid, putting the latter at 1-03
density, we get the following data for the mean density of the whole mass :

With density of solid crust 2-5 . . 2-2795
With density of solid crust 2-6 . . 23645
With density of solid crust 2-7 . . 24495

92

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS

Total difference, 0*17, or about 7-5 per cent, of the lowest value. Probably 2-4,
with an uncertainty of 4 per cent., may be assumed as approximately correct.
Yours, truly,

F. W. Clarke.

The arithmetical mean of the first six of the above estimates is 2 6 7, and if we
accept that as the mean density of the solid part of the Earth's crust, and assume, in
accordance with Professor Clarke's figures, that 0844 of the crust is solid and 01 56
liquid, then the mean density of the crust will be

2-67X0-844+ 1-03 X 0-156= 2-41

which agrees closely with Professor Clarke's own estimate.

But we must not forget the many facts which seem to indicate that the Earth's
crust is approximately in a state of hydrostatic equilibrium, and lead to the conclusion
that it is more dense beneath the ocean than in the continents.* If such is really the
case the comparative lightness of the waters of the sea must be at least partially com-
pensated for by the increased density of the strata upon which they rest, and the
average surface density of the Earth must exceed Professor Clarke's estimate. Our
present knowledge is too meager to warrant any very definite conclusion, but as the
continental surface density probably lies between 2-40 and 2*72, we may take the
mean of these numbers and attribute to it a probable error equal to half their differ-
ence. In that way we find

Surface density of the Earth = 2-56 ± o-i6 (284)

The phenomena of precession and nutation, and certain perturbations of the Moon,
enable us to determine two independent functions of the Earth's moments of inertia,
from the first of which the Earth's flattening could be found if the distribution of
density in its interior were known, while from the second it can be found without
that knowledge. In employing the first function Le Gexdre's law of the distribution
of density is usually assumed, and we have now to examine the process thus arising,
and the result to which it leads.

If we put A, B, and C for the three principal moments of inertia of the Earth,
A being the least and C the greatest, then the precessional moment of inertia will
be (2C — A — B)/2C. Some writers have called this "the terrestrial constant of
precession and nutation," or even " the precession constant," but that is certainly
objectionable, because these words have long been employed to designate the annual
motion of the vernal equinox, and their use for any other purpose can only lead to
needless confusion. Serret gives f

2C-A — B_ 2W

2C vn2

where m and n are respectively the sidereal angular velocities of the Earth about the

*286^,p. 364. t«3. P-324-

ON nil SOLAR PARALLAX AND ITS RELATED CONSTANTS

93

Sun, and about its own axis. Accordingly, with the Julian year as unit of time, and
the value of m from page 78

n/m — 366^249 983
2^-A - B

2C

w = 6-283 075 94

— 38861 OO7 K

(285)

or, if A and B are assumed to be equal, and n is expressed in seconds <»l arc

C-A

C

r= 0000 1 88 403 48 u

(286)

From (230), after substituting the numerical values of its A, B, C, and cos go0 for
the epoch 1850*0

« = + 1089640 i;£ — 4-071 361 5$ (287)

and by substituting (287) in (286)

C-A
C

— o-ooo 205 292 o^— o-ooo 767 058 7^1

(288)

Formula (286) is perfectly general, but (287) and (288) apply rigorously only
to the epoch 1850, and consequently $ and ^ must be reduced to that epoch before
being employed in them.

From (231), u— 17 34866", whence by (286)

C-A

= 0-003 268 55 =

C J 30595

(289)

For convenience of reference, some of the values which other investigators have
found for n and (C — A)/C are given in Table VIII. That attributed to La Place
is what he himself computed, but his values of the precession and nutation constants
are respectively 50-261" and 1006" when reduced to 18500, and their substitution in
(288) gives a somewhat greater result, namely (C — A)/C =10002 601 61". Bessel

Table VIII. — Values of h and (C — A)/ (J according to various Authors.

Date.

1799
1818

1830

1 841
1856

1859

1862
1889

Author.

La I'i.ai e (7, T. 2, liv. 5, chap. i. \\ 13-14)

! I ( I, ]>. I30) ...

Bessel ( i ^, pp. vand xv)

C. A. !■'. Peters (109, p. 161)

Le Verrier (8, T. 2, p. 174)

Serret (83, p. 323)

Hansen (55, p. 472)

Formulae (231) and (289)

18-312

17-362

«7'323
I7-378

(C-A):C

0-002 596 62 = 1
•002 924 50 = I
•OO345O l6=I
•OO3 27I I 2 1

•003 263 77 = I
OO3274 13=1
•003 272 = I
17349 I 0-00326855 = !

385"

341-93

289-84

305-7I
306-39
305-42
305-62

305 95

94

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

did not explain how he obtained the result quoted in Table VIII from the Funda-
nienta. The precession and nutation constants employed in that work are respectively
50*31 6 14" and 9/648 9", for the epoch of 1850*0, and their substitution in (288) would
give (C — A)/C r: 0002 928 28. The Tabulae Regiomontanae contains neither x nor
(C — A)/C, but its values of the precession and nutation constants are respectively
5Q'35i 36" and 8*977 97" f°r l850*o, and their substitution in (287) and (288) leads to
the values given in Table VIII. Le Verrier gave only h, from which (C — A)/C has
been computed by means of formula (286).

Taking the polar semi-diameter of the Earth for unity, if we putft for the polar
semi-axis of any one of the hypothetical equipotential strata composing the Earth, and
p and e for the density and ellipticity of the same stratum, then Clairaut's derived
equation may be written*

° = ( v w - 6£)l>™ + <» % + ■ y («*»

and from its integration the theoretical relations of the quantities discussed above will
result. But in order to effect the integration it is first necessary to express p in terms
of b ; or, in other words, the law governing the distribution of density in the interior
of the Earth is required. Le Gendre's (commonly called La Place's) law has usually
been assumed, and the investigation has been put in various forms by different authors.*!
We shall employ the expressions given by Thomson and Tait,"!" viz.

(291)

C -i-6(/-i)//«92

whore /is the ratio of the Earth's mean density to its surface density; 60 is the ratio
of the centrifugal force to gravity, both taken at the equator; and e is the Earth's
flattening.

These formulae afford the means of deriving e and/ from the observed value of
(C — A)/C ; but here we encounter the difficulty that 0 is the real independent varia-
ble, and any attempt to change it leads to very complicated algebraic expressions.
To avoid them Table IX has been formed, in which the numerical values of all the
quantities involved in the equations (291) are exhibited throughout a sufficient range
of the argument 0. In deriving e from SGa/2e w© have taken

». = *£= 0003 467 833 = 2-gg^ 09*)

where t^— 7424 252 0688, a — 20926 202 feet, /-= 3*251 169 feet, from (3), (13), and
(17)-

*28l, p. 276 and 321, vol. I, pp. 225, 226.

f294, p. 408; 321, vol. 2, p. 117; 7,T. 2,liv. 5, chap. 1, \ 14 and T. 5, liv. n, chap. 2, g 6; io, T. 2, p. 472; 17, p. 235;
24 >£ ; 15 j£, pp. 1 1 1 and 149 ; 13, pp. 83-87 ; 14, Teil 2, p. 487.
J II, vol. I, part 2, pp. 407 and 414.

/=

•3 (I-

- 9 cot 6)

5fo _

2€

/e2

'3(/-

3

0 /

c

-A_

£

— i^o

ON THE SOLAR PARALLAX AND Ms RELATED CONSTANTS.

95

Entering Table IX with the observed value of (C — A)/C from (289), viz,
o#oo3 268 60, we find

0=I44'6529° /= 2145 96
£ — OOO3 359 4 — I : 29767

(293)

Whence, with the value of the Earth's mean density from (283)

Surface density of the Earth z= 5-576/2-146 = 2-598 (294)

Table IX.— Numerical Values of the Quantities which enter the Equations (291).

-?i8o°

-

(i

/

5<r<>

2e

t

C-
C

A

1 34°

2338 74

1787 20

2460 73

1 : 283835 =

0-003523 17

I : 28885 1 =

0-003 4'' ' 99

'35

2356 19

1-81356

2 47O 87

I : 285 004 =

0003 508 72

I : 290- 259=:

0-003 445 20

136

373 65

•841 19

•481 22

286- 198

49408

291703

428 14

'37

•391 «o

•870 14

•491 79

287-418

479 25

293186

41080

138

408 55

•90048

•502 59

288664

46424

294-708

393 19

'39

•42601

•932 29

513 63

289936

449 04

29627 1

375 29

140

2443 46

196569

2-52490

I : 291237 =

0003 433 63

I : 297-875 =

0003357 n

141

•46091

2-00080

•536 41

292-565

41804

299523

338 64

142

•478 37

•037 75

■548 18

293922

402 26

301-217

31987

'43

•495 82

•07671

•56021

295310

38627

302-957

30080

144

■51327

•11785

•572 50

296-727

37009

304746

281 42

'45

253073

2-161 36

2-585 06

I : 298176 =

0-003 353 72

1 : 306586 =

0-003 261 73

146

•548 '8

•207 46

•597 91

299658

337 '4

308-479

241 71

'47

•565 63

•25636

•6ll 05

301 174

32034

310428

221 36

148

58309

■308 32

624 49

302724

303 34

312-435

20067

149

■600 54

•363 60

•638 24

304310

28612

314-501

17964

150

2-617 99

2-422 49

265231

1 : 305933 =

0003 268 69

I : 316630 =

0003 158 26

Le Gendbe's law of the distribution of density within the Earth is given by the
equation*

p = ~ sin-= sin 0 (295)

K

where p is the density at the distance r from the Earth's center, F and u are constants,
and 0 — r/x. Putting/, p\ and & for the surface values of r, p, and 0, and taking r'
for unity, we have from (293), (294), and (295)

F = p'/sin Q' — 4-490 7

Further, if n is a fraction such that w/ = r, then nr' /k — n6'- and therefore, upon
substituting in it the numerical values of F and & , equation (295) takes the form

pzz:^9°^ sin (w 144-652 90)

(296)

n

*i 1, vol. i, part 2, p. 404.

g6 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

To evaluate p when w becomes zero, (295) must be written

P-

sin

r
F

«

whence, by differentiating- both numerator and denominator, and making r zero, we
find for the center of the Earth

P =

1 r

-cos

K K

I

F

u

---e'Y

(297)

Table X exhibits the values given by formulae (296) and (297) for the density of
the Earth at various distances from its center; distilled water at 392° Fahrenheit
being taken as unity.

TABLE X. — Density of the Interior of the Earth according to Le Gendre's Law.

Distance T.
, . Density,
from center. '

Distance
from center.

Density.

10
09
0-8
07
o-6
OS

2598
3812

5057
6-292

7473
8-558

04

o-3

0-2

01

OO

9506
10284
10862
11-217
ii-328

Upon comparing the values of the flattening and surface density in (293) and
(294) with the corresponding observed values in (4) and (284), viz:

€ = 0-003 407 5 = I : 293-47

Surface density of the Earth zz 2 56 ± 016

(298)

a satisfactory agreement is found only in the case of the surface density. The two
values of the flattening differ largely, but our knowledge respecting the figure of the
Earth is scarcely sufficient to render it certain that the discordance exceeds the pos-
sible effect of errors of observation. Although the value off in (293) does not agree
with that derived by General Clarke from his discussion of the great geodetic arcs, it
is nevertheless within the limits of the values found from pendulum experiments, as
will be evident from an examination of Table XI.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.
Table XL — Flattening of the Earth as found from Pendulum Experiments.

97

Date.

1S30

I.S.U

1842

1853
1S69

1876

18S0

1SS4

1S84

Author.

e

Airy (17,^231)

Ham .y (29, p. 94)

BORENEUS (29V, p. l8)

Paucker (25^,T. 13, p. 230)

UNFERDINGER (39, pp. 313, 324, and 329).

A. Fischer (24^, p. 87)

Clarke (13, p. 350)

Helmert (i4,Teil 2, p. 241)

Hill (S7'A> P- 339. foot-note) :

I : 28289
I : 285-26
I : 289-
I : 28838

i : 29915
I : 284*4

I : 2922 J- 15
I : 29926 -J- 126

I : 285-44
I : 29002

(C — A)/C is certainly much better known than e, and more data for the deter-
mination of the latter are greatly needed. Respecting its derivation from pendulum
experiments, we may remark that when the length, of the pendulum is expressed by a
complicated formula, such as Unferdixger's or Hill's, the simplest procedure will be
to compute the numerical length of the pendulum at the equator and at the pole, and
then, calling these lengths respectively l0 and Z^, formula (i i) gives

1 S, ioa\

When the pendulum formula gives different lengths at the two poles, different flatten-
ings will result for the two hemispheres. Perhaps something might be gained by using
Airy's extension of Clairaut's theorem to terms of the second order.*

If 4 is the mean density of the Earth; i, p', and e' the values assumed by b, p, and
e at the sea-level ; and p — /(6); then it can be shown that after integration (290) must
satisfy the following conditions :f

I

II

III

IV

/(0=P'

lbV(b)db c

-A

£

bY(b)db

zxJ

2/(0)-,

(i6^,p.562.

>)"

db

db

Cry - i<70)

5^0

(299)

t320>P-523.

€987-

q8 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

With our own numerical values from (4), (283), (284), and (292)

%4=i 859^0-0053
p' = 2-$6 ±0'i6

C-A

1-9530 (300)

f^= 0-39305

Roche,* Lipschitz,! Darwin,}: Tisserand,§ Hill,|| and LevyH have used forms of
f(b) other than Le Gendre's law for integrating equation (290), but, singularly enough,
when substituted in the left-hand member of IV of (299) they all give very nearly
1 -987, while with the geodetic value of e the right-hand member gives 1 -953. Poincare
has recently investigated the subject in a more general manner** and has concluded
that no form of /(&) which is continuous between the limits o and 1 can satisfy the
observed values of precession and nutation together with General Clarke's value of
the flattening. He adds incidentally that the limiting values of the left-hand member
of IV of (299) are 1-987 and 2 04 ; whence it follows that the limiting values of £ are
1 : 296 and 1 : 300. We are therefore in this dilemma: Either the flattening must
lie between 1 : 296 and 1 : 300, or the distribution of density within the Earth can not
be represented by any function which increases continuously from the surface to the
center.

Those perturbations of the Moon which arise from the figure of the Earth have
been discussed by many geometers, among whom La Place, Plana, Pontecoulant,
Hansen, and Hill are conspicuous. If we designate the maximum values of the per-
turbations in latitude and longitude respectively by <5,s and Sv, then according to HiLLff

Ss-/J-,G 6v-ftlB. (301)

«2- «,-

where

3 3 3 2 3 96

4 4 of 9 nrn J

+ (r\+y + 6e?+1*?-2 £AL-4f+I3f|3/ (30,)

V 4 2 Q xmrn m 18 288

-2oyW-e^-W + ^-S .*,)^

9

44
"52 4 "* 576 3456

4- 3?925 tf_!9«._ 3449 m_ 59245^2

Hi f 1 38 „ 3 20 , 0.1^2 yip \ 1

= +( +%- r — 7Y — — ye; - 1 9xer + -*- -V77 )— 2

V 3 v 3 9 nrn Jnr

(303)

*3U. t296and297. £286. £31731^319. H289. ^295. ^303, p. 67. tt57K-PP- 2I3> 308, and 316.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

99

With the numerical values of m,w", y, eu c,, i/\ and "_./", from (54), (33), (64), (24),
(61), (241), (168), and Table VI, (302) and (303) give

G = — 1 1977097 II — + 105-29095 (304)

Further

a(i+*') ^ = 3siu2-(c_A + li)

sin P 2E V 2 y

and by substituting these values in (301) we have, when A and B are assumed equal,
and Ss and Sv are expressed in seconds of arc

t, sin2 P sin 201 /ri . . Ss Sv , s

-(C-A) = - = — (305)

2EV(i + x')2arci" G "II

Whether the Earth was originally fluid or not, for our present purpose it may be
regarded as covered by a fluid, because all observations relating to its figure are
reduced to the sea-level; and by combining that fact with the single assumption that
its interior is composed of nearly spherical strata whose densities are any function
whatever of their distances from the center, La Place showed that*

2C — A — B = — 7r(e — i«r0)a2 f prhlr

f F"

whence, as prdr = — , and A and B are assumed to be equal

C-A = 2(e-i«r.)EV (306)

By eliminating C — A between (305) and (306), the following expressions result
for finding: the flattening of the Earth from the observed values of Ss and Sv :

• sin2Psin2GL> , , _ N Ss Sv

(a-l*d=- = — (307)

(i + «')2arci" G H

With the numerical values from Table VI, (57), (7%)i (292), (302) ana* (3°3)>
(305) and (307) give

C — A = — o-ooo i3444E'«2(5s /,ogx

— + 0000 152 93 E' arSv

e — + o-ooi 7^3 92 — o-ooo 201 67 Ss (^OQ)

— + o-ooi 72)0 92 + °'000 229 4° $v

On account of accidental errors in the observed values of Ss and Sv, the two
formula' in each of these pairs will seldom yield identical results. To obtain the most

*7, T. 3,liv. 7, chap. z,\ 20.
-- DEPARTMENT OF PHYSICS

Case School ot FipplicD Science

CLEVELAND OHIO -

IOO ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

probable values of C — A and e we must therefore resort to the method of least squares,
and when the probable errors of Ss and Sv are equal, that gives

C — A = ( — o-ooo 075 836 Ss + 0000 066 668 Sv)E'a2

£=r + o-oo 1 733 92 — 0000 1 13 75 Ss-\- o-ooo 100 00 Sv '

The formulae (309) are independent of all theories respecting- the distribution of
matter within the Earth, Sir Geo. G. Stokes having shown* that equation (306)
remains valid, whatever that distribution may be; and, as Dr. Hill has carried the
expressions (302) and (303) to terms of the eighth order in the lunar theory, the
uncertainty in the coefficients of Ss and Sv is not likely to exceed four or five units in
the last place of decimals.

For comparison with (309), it will suffice to quote the formulae found respectively
by Pontecoulant and Hansen. Those of Pontecoulant are algebraically identical
with (307), but his series for G and H are not pushed so far as in (302) and (303), the
expressions employed being f

G = (_2_w-i§^ + ei* + 4tan'iy-2

V 3 4 18 3 Sm2

tt f . T9 . *3 \tanl

V 3 8 y m*

With our values of the constants, from (24), (54), and (65), these expressions give
G = — 1 2 1448 40 H = + 103*88052

whence, by substitution in (307)

£=: + °'001 J2>2> 92 — o*ooo 198 88 Ss
r=-|-o-ooi 7 t,t, 92 -f o-ooo 232 51 Sv

(3")

(312)

Hansen's formulae are, in his own notation!

^yi+4 shr-U" a + rj—p

p=:£(p — 2( — ) — — ; -A,

vL)y sin e cos e 2 + (a + V — i;)

^=20 siniJ^i +|sin21J^i — £0+ T?~ P)~\ Ai

His a, 77, and^j are related to our «/', //, and ipu through the expressions

«/'(> + -7) = >" »"P = A

and as his p, g>, D, J, e, A1} and ttu are our e, ff0, 1\, I, 00, Ss, and Sv, the formulae {312)
become in our notation

e=^0-i±±^[n2^avcl" . *-** Ss

sin2 P sin 100 211I' -\- n — tpx (7,17,)

Sv - 20 sin £I(i +^ sin2 £I)[i - h(n — A)/<]Ss

2

*&Vz, p. 680. fio, T. 4, pp. 485-4S6. J5S, pp. 348, 469, and 471.

ONTIIESOI.AU PARALLAX AND IIS RELAT1 D CONS! Wis

IOI

With the numerical values from Table VI, (49), (53), (65), (78), (241) and (292),
(313) gives

Sv — — 0*898 624.6s
£ = + o-ooi 73s 92 — o'ooo 196 62 6s (3*4)

= -f o'ooi 733 92 +0000 218 80 ^k

The formulae (309) and (311) are directly comparable with each other, but not
with (314)- Owing to the peculiar form of Hansen's lunar theory, his perturbations
differ from those found in the usual ecliptic .theory, and to avoid the troublesome
transformations which occur in passing from the one theory to the other, we shall
derive the necessary transition factors by comparing (309) and (314). Distinguishing
Hansen's 6s and 6v by accents, and equating the corresponding expressions in the two
sets of formula?, we have

201 667 6s •=. 196 61 7 6s'
229 400 6 v = 2 1 8 798 6v'

Whence

6s = 0974 96 6s' 6v = 0-953 78 Sv' (315)

Table XII contains the values of 6s and 6v determined from observation by
various astronomers, together with the resulting values of C — A and e, computed by
means of the formulae (308), (309), and (310). The values of 6s and<5^, which Burg
originally derived in 1806, and revised in 1823-182 5, are based upon 3233 observa-
tions of the Moon, made by Maskelyne, at Greenwich, during the }Tears 1 765 to 1 793 *
Burckhardt's values are based upon "more than four thousand observations," which
La Place says were those of Bradley and Maskelyne.t Aiey's value is based upon
the entire series of lunar observations made at Greenwich during the years 1750 to
1 85 1, in which the data used by Burg and Burckhardt are included, and constitute
but a small part. Hansen never explained the derivation of his values, but simply
wrote : " Die Mondbeobachtungen haben mir gegeben Ax rr — 8-382", irl — — 7-624",
whence the numbers in Table XII result through the formulae (315):

// »

Table XII. — Observed Values of certain Perturbations of the Moon which depend upon the Figure of
the Earth, together with the resulting Values o/O — A and e.

Date.

1806
1812
1823
1825
1861
1862

Authority.

BLvRO (39 yz, Introduction Tab. de la Lune)
Rurckharot (45X> Introduction) . . .

1!'K<; (45K. s. 324)

Burg (45^,5.14)

Airy (44, p. 12)

Hansen (55, p. 470)

6s

rr
80

80

86

8172

Sv

n
6-8
70

729
6-44

+ 7-272

C — A

o-ooi 06003 EV-
000 1 073 37 V.'tf-

OOOI I38 21' I

o-ooo 984 87 Y.'.i'-
o-ooi 104 54 I "

0-003 323 92 = 1 : 3°°-85
0-003 343 92 = 1 : 29905

0-003 441 17 = I : 290-60

0003 211 26= I : 31140
0003 390 68 I : 294-93

*45/4.col. 314.

t57%> P- 225. La Place was a member of the Commission appointed by the French Bureau of Longitudes to examine
Burckhardt's tables, previous to their official adoption.

102 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Of the results contained in Table XII, those found from the data of Hansen are
probably the most trustworthy. Accordingly we shall take

C — A = o-ooi 104 54EV (3T6)

err 0-003 39068= 1 : 294-93 (317)

whence, by comparing (289) and (316)

A = 0-336 83 EV C = 0-337 93 EV (318)

where E' is the mass of the Earth, and a its equatorial semi-diameter.

This method of determining the numerical values of the Earth's moments of
inertia is due to Hansen,* who has pointed out that it affords a fine confirmation of
the Earth's increase of density from its surface to its center, the value of C for a homo-
geneous spheroid being much greater, viz, 0400 00 E'er.

25.— UNCERTAINTY IN THE VALUE OF e, AND ITS EFFECT UPON THE OTHER

CONSTANTS.

The actual state of our knowledge respecting the flattening of the Earth is best
shown by bringing together the more important determinations of that quantity from
(4), Table XI, (293) and (3 1 7). They are

From geodetic arcs (Clarke) 1 : 293-47

From pendulum experiments (Hill) 1 : 287-71

From pendulum experiments (Clarke) 1:2922

From pendulum experiments (Helmert) 1 : 299-26

From precession and nutation, combined with Le Gendre's law of density . 1 : 297-67

From perturbations of the Moon 1 : 29493

Most unfortunately, every one of these results is affected by much uncertainty.
General Clarke used six separate arcs in determining his spheroid of 1880, all of which
were well represented by equations (3) and (4). Recently two of the longest of these
arcs have been connected, namely, the Anglo-French and the Russian, and according
to statements made at the Paris International Geodetic Congress of October, 1889, it
now appears that the similar ellipses passing through them have neither a common
center nor common axes! Thus General Clarke's e is invalidated, and the possibility
of deriving a trustworthy value from the present arcs becomes questionable. The pen-
dulum experiments are equally unsatisfactory. The result which Helmert deduced
from them by the condensation method, differs largely from that found by Hill through
a formula involving twenty unknown quantities, while General Clarke's value, com-
puted yet otherwise, agrees with neither Helmert's nor Hill's, but lies midway
between them. In 1827 Biot discussed a number of pendulum experiments, which he
divided into three groups, having their respective centers as nearly as possible at the

*287yz,s.i9s.

ON 111!*; SOLAR PARALLAX AND ITS RELATED CONS1 WIS.

' ' »3

equator, at latitude 450, and at the pole. From the first and second groups he found
err I : 27638 ; from the second and third, t — 1 : 306*33 ; and from the first and third,
e_ r •. 2ox>-59.* These results look as if the flattening varied with the region con-
taining the dominating number of pendulum experiments, but it is more probable that
their discordance arises from accidental peculiarities of the stations occupied, and there
is reason to fear that even yet we have not accumulated sufficient data to eliminate all
such peculiarities. As to the result derived from precession and nutation, it is evi-
dently vitiated by our ignorance of the internal constitution of the Earth; and even
the theoretically exact value from lunar perturbations is rendered questionable by tin-
uncertainty attaching to the observed values of the perturbations themselves. Indeed
the facts thus far adduced scarcely warrant any conclusion more definite than that
the flattening probably lies between 1 : 290 and 1 : 300, but we shall see presently that
there is some further evidence which tends in the direction of the smaller limit.

We have next to examine how much the system of constants given on page jt, is
affected by the uncertainty in the Earth's flattening, and on account of the number of
variables involved, the simplest procedure will be to recompute them all with an
assumed flattening of 1 : 300. For that purpose the numerical coefficients in formula-
(156), (157), and (166) require modification.

A comparison of (152) and (156) shows that the logarithmic coefficient of the
latter is

X\og 4«(i-M)3 4-68557487 = 278499322 (319)

Here I, p, and a are functions of e , and when e becomes e + de, I, p, and a become
respectively I + dl, p + dp, and a + da. Accordingly, we may write

(l + dl){p + dpf [1+ {a + da) V§] lp\i+Gs/%) l + dl (p + dpf

w l + gvi

and, therefore,

A log 4«(i + »)3 -4-68557487

* *(*+rf9(p + #mi+(* + Ar)Vt] 0 (32o)

= 2784993 « + ilog^ + ilog— ^— + 41o"

>g

l + dl J (p + d/o)2 i + 0 + ^)Vg

Formula (11) gives for the length of the seconds pendulum at latitude 45 °

li5 = k(i — _0 + *oa/2t{

whence

I _^— io«/2*i2 (32i)
° " i-2f -

. - - . . - — ■ '

* 20^, P. 38.

104 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Again, as arc sin V& = 35° 15' 52", the length of the pendulum at that latitude is

\oa_c-. ioa\*—fc 10a . .

l« = : 4(i - JO + w - (/■ - ^rj rrp + ^7 (322)

or, with the numerical values of li5, a and tx from page 9

?353 = 3'245 629 7 * + 0009 395 4 feet (323)

I ££

Differentiating (323), and dropping the subscript 35*3

di_ 3-24563 rf Q-54Q 94 de ,.2.s

(U- ecl^io5 (T=^y2^ (324)

and then, with sufficient accuracy

ri--i-^=i-|°-54Q94de
Z + dZ 2 ^1— if)

lo». l __ Q-54Q 94 M,

where M is the modulus of the common system of logarithms.
From (4) and (5)

and with that value of e2, and sin 2q> = £, (7) gives

Differentiating

(325)

e2zr2£ — e2 de=z± 1 — (326)

0 1 — as 4- 6e2 — 4«3+ £4 / _v

P=- -, „,^ (327)

3 — 2 £ + £

dp = - 6 + 3Q' ~ 44*2 + 28e* - 10^ + 2^ ( g)

and then, with sufficient accuracy

P2 _ p2 _ _ 2dp

(P + <fo)2 ~ ^2/ 1 + 2(fox ~ "p"

-!■■! 6-3Q£ + 44£2-28£3rff (329)

9 — 1 8f + 29c2 — 36e3
log P2 - ■ _l6-30^ + 44£2-28£3m.

The differentiation of (148) with respect to (J, 1, and e gives

*=«[ "f> de-fl ■ (330)

( 1 — <r sin'1 #> Z >

ON Till': SOLAR PARALLAX AND ITS RELATED CONSTANTS. 105

and by substituting the values of dl and de from (324) and (326), together with
sin2 cp — £

<fo = <MLZUg Q-54Q94 de ( 0

Also, with sufficient accuracy

i+<V§ __ _ da^'i

1 +(* + **)<•$ i+<V§

and therefore

log ' + g^ <xMy§ ( im.. 054094 J (h. , 2)

With M zz 0-434 294, and £ zz 1/293-466, e2zz o'oo6 803 48, Zzz 3 '256 87 feet,
<TzzO-oo2 829 7, 1 +<V§ = 1*00231, from (4), (5), (14), and (154), (325), (329), and
(332) give

^T^Jl + ^^fW' + i '°g , +'(*+&} (333)

zz J(— C072 379 + 0-286 564 — cooo i66)cfe

A comparison of (152) with (157) shows that the logarithmic coefficient of the
latter is

log- — 4«( ! + * ) — _ 4-665 070 71 — 10

whence, by reasoning identical with that employed in connection with (319)

los: — ' H zz 4*66^070 71 +0-21402 de— 10 (^4)

8a + dO(P + rfp),T1*[i+(ff + d(F)v§]

Upon equating (157) and (165) we obtain an expression of the form

(335)
--^-sin3Pl
AB B

where, from (334) and (l65)

i/Azz 216 236-65— 106 561 de Bzz 10288642 CZZ38442769

and the substitution of these values gives the third formula of (336).

Collecting our results, we have from (156), (157), (166), (320), (333), (334), and

(335) „/ = V

p zz [2-784 993 22 + 0-071 34 de]l — -J

E_ sin^P (336)

M [4-66507071 + 0-214 02 de— 10] — siiVP

/ 1 — (2 1 6 236-65 — 106 56ijfe)_sin3 P )

^=<^\ 3757 444 9 - o-o 1 o 357^— (80795 2 <64 - 398 1 57 ^f) sm' p >

106 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

These formulae are adapted to any value of the flattening between the limits i : 290
and 1 : 300, and in order to facilitate their use the following supplementary expressions
are added :

dezz e — 1/293466 =. £ — o'ocv} 407 55
log (216236648— 106 561 de) = 5-334929300 — 0-214 020 de , .

log (3757 444 92 — o-oio 357 de) — 0-574 892 623 — o-ooi 197 de
log (807952643— 398 1 57 de) — 5907 385 906 — 0-214021 de

With a flattening of 1 : 300, de — — 0*000074 22, and the conditions which must
be satisfied by the adjusted quantities are, from (336), (175), (180), (182), and (18

1 sin3 P

M [4665 054 83 — 10J — sin3 P
o or r, -p- [2784 988] f 7qrjj\

oorv2=p — [5303 125 — io]PQ^^j

0 or v3=p — [4681 962— 10] PL1^"— (338)

[8912482]
o or t>4 = i> — L yV(^ — J

[7-526016]
O or i'5 rz « — L— ■ „ ^-J
\ «

oorP,=i-gj . - 216 244-56 sin3 P>

*< 3-757 445 7 — 807 982-20 sin3 P>

The values of the observed quantities will be the same as in (195); except those
of P and <§, for which (70), (75), and (204) give

P = 3 423-08"+ 5062"^ ±0-121" .

g= 50-358 6"- 31 716" dE ±0-00248" Uo9;

With the above value of de, and d E = + 0000 000 05 1 , we therefore have

p — 8-834" ± o-ooS 6"

P=r3422704 ±0121

g = 50-3570 + 0-00248

<& = 9-233 1 +001 1 2

Q= 125-46 ±035 ( o)

L- 6-514 +0016 ^ '

a — 20-466 +o-Ol 1

6 rr 4970" + ro2s

V = 186347 + 36 miles

E rr 0*000003 °°5 °97 ±0-000000016056

.

ON THE SOLAR PARALLAX AND lis RELATED CONSTANTS.

'"7

The substitution of these observed values in the conditional equations (338) leads
to the following- system of numbers

1 /M = 81722 6

Vj = + 0-073 81" #4 = + 0007 20"

v2 — — 000959 Vry= + 0-029 99 (341)

'V. = — 0033 55 i'c = + 0-035 60

from which the corrections by adjustment given in Table XIII result by a double
application of the formula1 (193), precisely as in the case of Table VI.

It is now desirable to have a method more direct than that employed on page
72 for finding the probable errors of the masses of Mercury and Venus after they
have been corrected on account of dE. As the expressions (203) are of the form
A' + B'rfE — (1 + v)/mo, the expressions for v, v\ and v" will be of the form
v — A"-|-B"cZE, and when they are substituted in the observation equations (129)
the resulting residuals will be of the form, v — A'" + B'VZE. We may therefore write

^y = A + B.rfE + C(dE)2 (342)

and if the probable error of dE is =h SE', when that quantity is given there will be an
additional term of the form + D(rfE')2. The algebraic expressions for A, B, C and D
are simple enough, but they are not needed here because the numerical values of these
constants can be most readily found by an indirect process.

The residuals on page 48 give 2vv zz 4*257 31 ; those on page 72 give, for
rfE = + 5 1 440 =t: 27 33 1, 2w = 5-500 65 + 0-578 55, where 0-578 55 is the part aris-
ing from the probable error of dE ; and a special computation gives, for dE zz + 25 000,
■2^ = 4-551 04. In accordance with (342), these numbers yield the equations

4-25731 zzA

5-500 65 zz A + 5-144 o B + 26-460 74 C ^ ^

4-55 1 04 = A + 2-5000 B+ 6-250 00 C

0-57855 = 7-469840

where rfE andtfE' have been multiplied by 100 000 000 for convenience in printing.
The solution of (343) gives

^ = 4-25731 + [3627 3>7E + [14-671 gi](dE)- , .

+ [14-889 o3](^E')2 V°44;

the quantities within brackets being the logarithms of the numbers they represent.

From (198), (203), and (344), together with the usual formula for probable error,
the following general expressions for the corrected masses of Mercury, Venus and
the Earth have been derived :

„ „ ,, 0^44647 + 266 200 dE ± R
Mass of Mercury zz J-

J 3 000 000

Maas of Venus =<yv>*196 - «*«>7 <M±« (345)

40I 847

ro66 618 + 354 936 (dE±dE')

Mass of Earth zz

354 936

io8

on the solar parallax AND its related constants.

where dE is the correction to the mass of the Earth given by (193), (IE' its probable

error, and

Log R =%\\og.2vv + 6-979 147}
Log R' =: \ \ log. 2vv -f- 4'566 211}

(346)

The probable errors in the second and fourth columns of Table XIII have been
computed from the data in Table V, with q— 1375 8 ; and the values of the appended
planetary masses, together with their probable errors, have been computed by means
of the formulae (344) to (346), thus avoiding the laborious process described on
page 72.

Table X [II. — Results for the Epoch 1850-0, uponthe assumption that the Earth's Flattening is 1 : 300-00.

Quantities.

Observed values.

Corrections by ad-
justment.

Adjusted values.

// //

//

// //

/

8-834 + 001 1 83

— 0-02463

8-80937 + 0-00554

P

3422-704 + 0-16647

— 0-10774

3422-59626 + 0-03897

1

50-3570-!- 000341

4- o-ooo 1 2

50357 12 + 0-00341

1

9-2331-!- 0015 41

— 0-013 J6

9-21994 + 0-00823

Q

12546 + 048153

— 0-505 38

124-95462 + 0-08003

L

6514 + 002201

4- 0-00740

6-521 40 + 001804

a

20-466 + 0-015 l3

— 001 1 35

20-45465 + 0-01236

e

497-o» -J- 1-40332'

+ 1-009 8i»

498009 8i8+ 0-30067'

V

186347 -J- 49-529 miles

— 10009 miles

186336-99 +48-542 miles

E

o-ooo 003 005 097
-J- o-ooo 000 022 090

} +51-1840 {

0-000003056281
+ 0-000000005 784

M

0-01233207 + 0-00003458

Mass of Mercury = °'358 2?2 ± °^2 ^ = .

3000000 8373526^1766213

Mass of Venus = °'^2 ^9 ** f00^ 5°? = 1

401 847 408 984 ± 1 876

Mass of Earth = Ho84 ?85 ±0*002 053 _

Mass of Moon —

354 936
1

327 i95±6i9

81*089 4 ±0*2274

Mean distance from Earth to Sun = 92 797 600 ± 58 318 miles.
Mean distance from Earth to Moon — 238 861 ± 2*719 miles.

As the correction which would accrue to the equatorial diameter of the Earth
upon changing the flattening from 1 : 293*5 to T : 3°° i-s °f the same order as the prob-
able uncertainty in the diameter itself, the distances of the Sun and Moon appended
to Table XIII have been computed with the same equatorial radius as those appended
to Table VI. Nevertheless, it is important to inquire how a small change in the
flattening will affect our adopted values of the Earth's equatorial and polar semi-

i__j

ON THE SOLAR PARALLAX AND [TS RELATED l ONSTAN

IO9

diameters, and we therefore revert to General Clarke's final equations for the deter-
mination of these quantities. They are*

0 = + 5666 1 5 + 301762 4.11 + 126925 2V
o — — 16-967 7 4- 126-925 2ii-\- 22 1 430 7 v

(347)

and our first task will be to express u and v in terms of db and de. To that end, let

b

- ■=. a0 4- da

I — £

b = b0 + db
b0=z 20855 500 feet

Thenf

f — f0 + de

b = b0 + db = b0\i+— *__{

( 10 000)

1 . //

-= \- iov arc 1

n =

and from (349)
Further, as

a — b _

a-\-b 2 — e 590

u rr OOOO 479 490 db

(348)

(349)
(350)

(35 0

we may write £0/(2 — fo) = I/59Q> an^ from that, together with (350) and (352)

i^

+

*o (2 — c0)

€0= 1/295-5=0-003384095

de=. 10 v arc 1"

Vrr 10 348*2 de

(353)

The substitution of the values of u and v from (351) and (353) in the first equation

°f (347) gives

db-- 391-6 -9 077 539 de (354)

and by differentiating the expression «0(i — e0) = bd, and substituting the values of b0, e0,
and db from (348), (353). and (354)

da = -^L. -\--Ml^--39r9+ 11 SS9 01 1 de
i—e0 ( 1 — £0)-

(
Then (348), (353), (354), and (355) give

a = b0/(i —e0)-\-da= 20925923-7+ 11 889 on de feet
b — b0-\-db =20855108-4— 9077 539 de feet

and by changing e0 from 1/295-5 to r/293'47> we obtain finally

a — 20926 202 4- 1 1 88901 1 de feet
6 = 20854 895— 9 077 539 df feet

(355)

(356)

■i.P-3'7-

|i3,pp. in and 313.

IIO ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

For a flattening of i : 300, (337) gives <fc = — 0-000074 22, and by (356) the cor-
responding equatorial and polar seniidiameters of the Earth are

0=20925320 feet , v

h — 20 855 569 feet V^"

Upon combining this new value of a with the adjusted parallaxes contained in
Table XIII we find

Mean distance from Earth to Sun = 92 793 663 ±58318 miles (-irg\

Mean distance from Earth to Moon= 238 851 ± 2719 miles

and these results are substantially the same as those appended to Table XIII, the dis-
cordance amounting to ten miles in the case of the Moon, and to only one-fifteenth of
the probable error in the case of the Sun.

The most striking feature elicited by comparing Tables VI and XIII is the great
diminution of the probable errors in the latter ; and as that arises solely from the
change in the flattening, and extends not only to the quantities in the table, but also
to the appended masses derived from the equations (1 29), it seems desirable to inquire
what value of the flattening would reduce these probable errors to a minimum. A rig-
orous solution of that question would be very laborious, but a fair approximation may
be got by finding what value of e would make the residuals in (185) a minimum. If v'
and v" are the two values of any one of these residuals, given respectively in (196),
(206), and (341), and corresponding to the flattenings e' and e", then the condition that
that particular residual shall disappear may be written

o = v' + ~^r,de (359)

c c

and the resulting value of the flattening will be e' -\-de. The residuals v4 and v5 are
independent of the flattening, but vlf v2, v3, and v6 furnish the four observation equations

0 = + 0072 86"— i2'8"de
o = — 0-00532 + 57-5 de
o = — 0-25222 — 2946-6 de
o=z + oio6 66 -f- 957-5 de

from which we find, by the method of least squares

e'4-de = - — — 0-00008700=: (360)

293-466 ' 301-24

It is not easy to decide how much weight should be attributed to this method of
estimating e, but the result certainly tends to strengthen Helmert's value from pen-
dulum experiments, and our own from precession and nutation.

Doubtless more data are the first requisite for a satisfactory determination of the
Earth's flattening, but in default thereof one resource yet remains, and we shall avail

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. I I I

ourselves of it in Section 27. Meanwhile a symbolic correction may be applied t"
General Clarke's geodetic value, by writing

e= 1/293-466 + de (361)

and then every quantity in, and appended to, Table VI will take the form

x = x' + ^X"llde (362)

£ — £

where x is any one of the quantities in question, x' and x" are respectively the
numerical values given for it in Tables VI and XIII, and e' and e" are the values
of the flattening with which these tables were computed. As e' — 1/293-466 3, and
e" = 1/300, equation (362) reduces to

x = af+ 13475 (at — x")de (363)

The resulting numerical expressions are appended, and by their aid the values
of all the quantities in question can readily be found for any possible value of the
flattening.

d£—e—i/ 293-466 3 = e — 0-003 407 546
p- 8-80976"+ 5-26" <te±( 0-00674"+ 16-17" <fe)
P —3422-65643" + 810-79"^+ ( 0-04746"+ 114-40" de)
V = 50-357 15" + 0-40" de ± ( 0-004 15" + 9 '97" de)

Z\ — 9-21621"— 50-26" rf£± ( 0-01002"+ 24-I2"^)

Q = 1 24-962 05" + 1 oo- 1 2" de ± ( 0097 46" + 234-87" de)
L = 6-51303"— 1 1 2-79" de±( 002197"+ 52-g6"de)

a_ 20-45385"- 10-78" rfe±( 0-01506"+ 36-38"^)
0 = 497990 949 - 25427s <te± ( 0-366 168 +882-48" de)
V= 18633599 -13 475 de± (59-117 +142498^)
Mass of Mercury =

0-000000119477 ) (0-000000024220
+ 0-0000003 10 de 5 I + 0-000000 732 <?e (364)

Mass of Venus =

0-000002444953 > . Jo-oooooooii 241
— 0-000001738^) ( + 0-000000335^

Mass of Earth =

0-000003056537 > (0000000007043
+ 0000003 423 de) I +0-000016965^

Mass of Moon —

0-012315 7 — 0-22 10 d£+ (0-000042 11 +o-ioi 5rt£)

Mean distance from Earth to Sun —

92 793 504-54 857000^ ± (70993 +170 796 000 dt) miles
Mean distance from Earth to Moon —

238 g 5 7 - 5 3 900 de ± (3-3 1 2 + 7 99 1 d£) miles-

112 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

26.— MASS OF THE MOON FEOM OBSERVATIONS OF THE TIDES.

The first determination of the Moon's mass was made by Newton in 1687, from
the tides, and other investigators have since employed the same method, but for more
than 1 80 years it yielded no trustworthy result. Its failure was due to various causes,
both theoretical and practical, and although some of these were cleared up by La Place
as early as 18 18, there was little prospect of success until the recent application of
harmonic analysis to the reduction of continuous observations of the tides, recorded
by automatic gauges, and extending over long periods of time. An effort has been
made to collect in the following list all the more important determinations of the
Moon's mass which are based upon the method in question, but owing to the singular
sparseness of bibliographic references in writings on that subject, it is difficult to esti-
mate what degree of completeness has been attained.

Reciprocals of the Moon's Mass, determined from Observations of the Tides.

1687. Newton, from the tides before the mouth of the river Avon, three miles below Bristol, England. (9, lib. 3,

prop. 37, cor. 4.) 398

1738. D,. Bernoulli. (5, p. 549.) 70-

1818. La Place, from the tides at Brest, France, observed during the years 1807 to 1814, inclusive. (348, p. 55.) . . 69-3

1824. La Place, from a rediscussion of the above observations at Brest. (7, T. 5, liv. 13, chap. 3, \ 10.) 74-9

1831. Lubbock, from the tides at London, England, observed during the years 1808 to 1826, inclusive. (350, i83i,p.

392) 667

1854. Haughton, from the semi-diurnal tides on the coast of Ireland, observed during the year 1851. (344, p. 130.) . 63-

From the diurnal tides, during the same period. (344, p. 130.) 95-

i860. Lubbock remarks in his paper on the lunar theory (Mem. Roy. Ast. Soc. 1861, vol. 30, p. 29) that the observations

of the tides which he employed gave him a value "probably about 1:67-3 " f°r tne ratio of the mass of the

Moon to the sum of the masses of the Earth and Moon. His value of the reciprocal of the Moon's mass must

therefore have been 66-3, which agrees with what he found from the tides at London, in 1831, so closely as to

render it probable that they are the tides referred to.
1862. Haughton, from the semi-diurnal tides at Port Leopold, North Somerset, observed November, 1848, to July, 1849,

inclusive. (345,1866^.655.) s 65-4

From the diurnal tides during the same period. (345, 1863^.253.) 85-

1866. Haughton, from the semi-diurnal tides at Frederiksdal, Greenland, observed August, 1863, to August, 1864,

inclusive. (345, 1866, p. 642.) 64-6

1866. Haughton, from the semi-diurnal tides, observed at the following points on the coast of Ireland, during the year

1851. (346, p. 346.) :

Bunown 69-1

Cahirciveen 64-2

Castletownsend 49-2

Dunmore East 55-0

Courtown 107-2

Kingstown ... 46-3

Donaghadee 35-1

Cushendall 21-1

Portrush 846

Rathmullan 717

Mean 60-4

1867. Finlayson, from the mean range of the spring and neap tides at Dover, England, observed during the years 1861,

1864, 1865, and 1866. (343, p. 272.) 87-9

1870. FERREL, from observations of the tides at Boston, Mass., from July I, 1847, t0 July '. 1866. (334, p. 85.) . . . 78-6

1871. FERREL, from a rediscussion of the tides at Boston, Mass., July I, 1847, to July 1, 1866, with special reference to

the Moon's mass. (335, p. 198.) 75-1

1874. Ferrki., from observations of the tides at Brest, France, during the years 1812 to 1832, inclusive. (336, p. 189.) . 78-0
1874. Ferrel, from a second revision of his discussion of the tides observed at Boston, Mass., July I, 1847, to July ',

1866. (336, p. 196.) 817

ON nil SOLAR PARALLAX VND ITS RELATED CONSTAJJ ji?

1S74. Ferrel, from seven years' observations of the tides at Liverpool, England, 1857 to 1 860, and is

1'- -"')

T.S74. Ferrel, from observations of the tides at Portland breakwater, England, during the years 1851, 1S57, 1866, and

and 1S70. (336, p. 223.) 801

1S74. Ferrel, from observations of the tides at Fort I'oint, Cal., during the three years 1S5.S to 1S01. (336, p. 22^ 70 '.

1574. Ferkki., from observations of the tides at Karachi. India, during the three years 1S6S to 1S71. (336, p. 234.) . 7 S 1 .

1575. Ferrel, from observations of the tides made at Pulpit Cove, Penobscot Hay, Maine, during the six years 1870

to i875- (33s> P- 294) 833

1S82. Ferrel, from observations of the tides made at Port Townsend, Wash., during the three years 1874 to 1876.

(339.P-44S.1 772

1882. Ferrel, from observations of the tides made at Astoria, Oregon, during the three years 1874 to 1876. {syj, p.

443.) 681

j Ferrel, from observations of the tides made at San Diego, Cal., during the three years 1869 to 1871. (339, p.

448) 880

1SS3. Ferrel, from observations of the tides made at Sandy Hook, N. J., during the six years 1876 to 1881, inclusive.

(340, p. 251.) 771

Long ago Airy showed why the Moon's mass can not be accurately determined
from the mere ratio of the solar and lunar effects in the semi-mensual inequality of the
tides,* but nevertheless many of the values recorded above have been obtained in that
very way, and are therefore worthless. Those found by La Place's method, f or by
Ferrel's modification of it, are theoretically correct, at least for deep-water tides, but
instead of confining ourselves to them, we shall compute many new values from the
"Results of the harmonic analysis of tidal observations'' which have been published
by Major Baird and Professor Darwin.}:

La Place's method of deducing the Moon's mass is not adapted for use with har-
monic tidal constants. We shall therefore employ Ferrel's formulae, and as much of
the available data was collected by Sir William Thomson, Mr. Edward Roberts,
Major A. W. Baird, R. E. and Professor G. H. Darwin, the various notations adopted by
these gentlemen are exhibited in Table XIV. Ferrel used the letters A or a for the
semi-range, and e or e for the epoch of a tide, and distinguished the various classes of
components by the subscript suffixes i, 2,3, etc.; but for similar components occurring
in the diurnal and semi-diurnal tides he used the same symbols. Thomson used the
letters R and e, respectively, for the semi-range and epoch of any component, distin-
guishing the various classes of components by the initials S, M, O, K, etc., and indi-
cating their period by the subscript suffixes 1, 2, 3, etc.; 1 indicating a diurnal
component, 2 a semi-diurnal component, 3 a terdiurnal component, and so on. Darwin
used the same initials as Thomson to designate the various classes of tidal components,
but his H and u are not identical with Ferrel's A and e, and Thomson's R and e.
The semi-ranges and epochs of most of the tidal components are to a certain extent
functions of the longitude of the Moon's node, and are therefore subject to small
inequalities having a period of 18.6 years. The A's and f's of Ferrel and the R's
and e's of Thomson are affected by these inequalities, but the H's and jcs of Darwin
are very nearly free from them.§ In other words, the A's and e's of Ferrel and the R's
and e's of Thomson belong strictly to the years during which they were observed,
but the H's and h's of Darwin are reduced to what they would have been if the Moon

*323, p. 360*, art. 455.

f The basis of this method is very clearly explained by Airy, 323, pp. 360*, 379*, and 386* articles 455, 536, and 555.

t327-

§ Compare 353, 18S3, p. 86, and 338, p. 282.

6987 8

n4

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

had remained always in the celestial equator, and they therefore correspond to the
means of a series of observations extending over the entire period of 18.6 years,
except in so far as they may be vitiated by accidental errors. In explanation of Table
XIV it is only necessary to add that quantities on the same line have similar signifi-
cations. For example, Ferrel's diurnal A, and «?, correspond to Thomson's Hy and ex
of K,, and to Darwin's H and u of Kj; and similarly for the quantities on the other lines.

Table XIV. — Notation for Harmonic Analysis of the Tides.

Ferrel.

Thomson

and
Roberts.

Darwin

and
Baird.

Name of Tide.

Speed of Tide

per Mean Solar

Hour.

H„

Ao

Mian level of sea.

0
•

[A

A,

f|

K,

, R| «,

K, ,

H K

Luni -solar diurnal ....

15-0410686

c g

A,

O

. R. d

O ,

H K

Lunar diurnal

13943 035 6

5 B

A3

ea

P

, R, El

P ,

H K

Solar diurnal

14958 931 4

0

<?o

s,

, Ri ei

s, ,

H «

1st of principal solar series.

15-0000000

Ao

fo

M,

» R.

Ms ,

H K

2nd of principal lunar series .

28984 104 2

A,

fi

s.

, Rs 6,

S: ,

II K

2nd of principal solar series .

30-0000000

3 1)

A:

1

."

. Ra e.2

" 1

U K

27-968 208 4

= 5

A,

ft

K9

, R, e,

Ka .

II K

Luni-solar semi-diurnal . .

30-082 137 2

A4

*4

L

, Rj 6,

L ,

H *

Smaller lunar elliptic . . .

295284788

A6

N

, R. e,

N ,

H *

Larger lunar elliptic ....

28-439 729 6

"1

'1

M<

, K, e,

M4,

H k

4th of principal lunar series .

57-9682084

rt;

MS

, R4 *4

MS,

II x

Compound tide

58-984 104 2

> S

"3

s,

. K4 t4

S4 ,

H K

4th of principal solar series .

60-000 000 0

0 B,

5 £

«4

<'l

2SM

. K.. ea

2SM .

H *

31-0158958

<>,

M«

. Rg e6

M„ ,

H K

6th of principal lunar scries .

86-9523126

Our object is to determine the mass of the Moon, and as that quantity affects
only the ranges of the tidal components, and is without influence on their epochs, we
shall have to deal exclusively with the H's, and not at all with the k's. Our notation
may therefore be abridged by using the initials of the various tides to denote their
semi-ranges, and instead of H of S, H of M, H of O, etc., we shall write simply S, M,
O, etc.

For the diurnal tidal oscillations of the great oceans, Professor Ferrel's expres-

sions are*

Ki = A1 = (o-53o6— 13-1 <*//)( 1 -f 0-230 E^Ao

O = A, = 0-381 3(1 — 0-230 Ei) A0

P = A3 = (0-1730— 13-6 a»(i +0-196 Ex) A0

A4zz 0-084 0 +0-231 Ei) A0

A5 — 0-070 (1—0-231 E,) A0

(365)

•336, P. 89.

ON THE SOLAR PARALLAX AND ITS Kill. AMU CONSTANTS. 115

where A0, Ex, and Sju are constants to be determined from the values of A^ A . A . etc,
by the simultaneous solution of three or more of the equations. As the numerical
values of A4 and A5 are not contained in Baird and Darwin's list of harmonic tidal
constants, we shall neglect the last two equations of (365) ; but their loss is of little
consequence on account of their small weight By putting'

m — 13-6 Kt— 12-98 o — 131 P
2a =. 0-462 4 Kx + 5-529 O
6 = 0613 1 Kx 4- 0-585 1 0 — 0-693 o P

the remaining equations of (365) give

0 lb b'S

\ = (366)

0-381 3 (1- 0-230 EO VJ ;

8? = 0040 50 - 0-3813(1-0-230 EOK?

131 (i +0-230 E^O
Mass of Moon zz 0012 50 + 8/x

Although this solution is in the form best adapted to give exact numerical results,
it is too cuinberous for general use, and we take advantage of the smallness of Sju to
add another which is briefer and not appreciably less accurate :

6u — °'l22 3° Ki+o'oi3580— °4Q5 04 P
9614 36 K, + 1*513 63 O— 10'oooooP

Ex Ki — * > ' "391 56 - 3436 $M) _ P-0 (Q'453 7i — 35*7*m)

0230 Ki + 0 (0-3 20 06 — 7-90 Sju) o-23oP + 0 (0088 93 — 6-99 dpi) ^3 7)

A.= ° ■_

0-381 3(1 — 0-230 EO

Mass of Moon zco 1 2 50+^yw

For the semi-diurnal tidal oscillations of the great oceans, Professor Ferrel gives
the expressions*

S2/M2 = Rl= (0-458 2 - 36-2£//)( 1 + 0-425 5 E2)
p/M%— R.2=z: + 00240(1 —0-425 5 E2)

K,/M2 = K3 = (o-T256-3-2^)(i+o-4599E2) (368)

L/M2 = R4 - - 0-028 6(1+ 0-228 E,)
N/M, = R5 = + o-i92 2(1 -o-228E2)

The second and fourth of these equations may be dismissed at once on account
of their small weight, but there is a difficulty in deciding how the remaining three
should be treated in order to get from them the most probable values of E2 and dp.
The fundamental quantities M2, S2, K,, and N are not observed independently, but are
each functions of the same observed quantities, namely, a series of heights recorded
by a tide gauge, and therefore according to the theory of least squares it is the sum

*336. P-9i-

n6

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

of the squares of the corrections to the latter which should be made a minimum, sub-
ject to the conditions of (368). But even if these heights were at hand the applica-
bility of this process would be doubtful, because in the present case the larger errors
probably do not follow the usual law of error. However, as the only practicable
course is to deal with the derived functions M2 , S2 , etc., our discussion had better be
confined to them. If, on the one hand, we take the equations as they stand, and solve
by the method of least squares, the results are likely to be affected by systematic
errors, because all the absolute terms have been divided by the same fundamental
quantity, namely, M2 , which is itself subject to error ; and if, on the other hand, we
apply a symbolic correction to M2 , then we shall have as many unknown quantities as
there are equations, and the errors which should be distributed among- all the funda-
mental quantities will be thrown upon M2 alone. Under these circumstances it is
probably best to divide the third equation by the fifth, and thus we obtain the two

equations

So/M2 = B = (o-458 2 — 36-2^)(i +0-425 5E0)

K,/N = C = (°"653 5— i6-6o»(i +0-459 9 E2) (369)

I — 0-228E,

in which each of the fundamental quantities occurs once, and only once. Here, as in
(365), we give two solutions. The first is that which leads to the most exact numerical
results. For it we put

m — — 16-051 — 166B + 36" 2 C

2u — + 1421 1 + 7-634 3 B— 7-149 5C

^ = + 3-1409 + 3511 9C

and then

!4

E' = ~;±

Wl (f

B

S/x — o'oi 2 66

36-2 + 1 5-403 E2

Mass of Moon — O'oi 2 50 + 8/jl

(37o)

The second solution, which is briefer than the first and not appreciably less accu-
rate, is as follows :

§ _ 0-096 38 BC + 0127 04 B — 0-126 5 7 C — 0-004 354
3-236 72 B — io-ooo 00 C — 0-454 92

E 0 — 0-65349+ 1 6-65 8/x _B — 0-4582 + 36-2^

0-228 C +0 300 54 — j666ju 0-194 96 — 15-406/*
Mass of Moon — o*oi 2 50 + <5/*

(370

Professor Ferrel treats the equations (368) in a way which differs widely from
that adopted above, and which is best explained by the following- extract from his paper
on the tides of Penobscot Ba\

It is readily seen from an inspection of these ('([nations that they can be satisfied only very
imperfectly for Pulpit Cove, within any determined values of djx and E, and that they can be

* 338, P. 297.

ON THE SOLAR PARALLAX AND IIS RELATED CONSTANTS. 117

much better satisfied by multiplying the first members of the equations by an unknown constant.
This constant is introduced upon the hypothesis that the tidal components arc diminished by the
effect of frictiou which is as a higher power than the first power of the velocity, as I have at
various times explained. Upon this hypothesis large tides arc diminished by friction more than
small ones in proportion to their amplitudes, and hence where there is one large component, as
the mean lunar, and a number of much smaller ones, since the amplitudes of the latter are obtained
by analysis from the differences between the larger and smaller resultant tides, the smaller com-
ponents are diminished more than the larger ouesin proportion to their magnitudes, unless friction
is as the first power of the velocity.

Accordingly, Professor Feekel multiplies the Left-hand members of the first,
third, and fifth equations of (368) by a constant, which he calls c* thus reducing them

to the form

(S2/M2) c - RjC =r (0-458 2 — 36-2d»( 1 + 0-425 5 E2)
(K,,/M2)c=R3c = (o-i25 6— 3-26»(t +0-4599^,) (372;

(N / M2) c — R5c = o- 1 9 2 2 ( 1 — o- 2 2 8E2)

and he adds: "By the introduction of the constant c, or one equivalent to it, I have
in all cases found that the observations are better represented by theory, and a better
mass of the Moon is obtained, which indicates that there is an effect due to friction or
some other cause which diminishes the amplitude of the tides."

If we put

in — + 3-2 R,— 36-2 R3 + 16*027 R5

2« = — 0742 i Rj+ 7-1495 R3— i4-I9T Rs
b = + 0-335 5 Ki — 3"5i 1 9^3- 3,I36Rs

the general solution of (372) is

^ 0-192 2 — 0-043 82 E9
c = 1 — F =

dju = 0-012 66 —

Iw, (373)

Rx( 0-192 2 — 0-043 82 E2)
R5(36-2+ 1 5-403 Ej
Mass of Moon rz o o 1 2 50 -f Sju

or more conveniently, and with no real loss of accuracy,

„ _ 0-036 52 So— o- 1 26 58 K2 — o 004 35 N
~ 0-930 5 1 S2 — 1 o-ooo 00 K, — 0-454 92 N
F _ o- 1 92 20 S2 — N (0-458 20 — 36-2Qf>» . , ^

2 ~~ 0-043 82 S2 + N (0*194 96 — 1 5-4oo»
_ _ F _ (o- 1 92 20-- 0-043 82 E2) M2

Mass of Moon zr o 01 2 50 + o>

The last solution shows that Professor Ferrel's process results in replacing i/Ma
by an indeterminate, c/M2; and thus, instead of having four fundamental quantities

This c is identical with the i -F of equations (130), (345), and (354) of his Tidal Researches. (336, pp. 75- '88, and 195.)

n8

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

from which to find two unknowns, he has but three fundamental quantities from which
to find three unknowns.

Table XV contains a series of corrections to the Moon's mass, deduced from tidal
observations made in thirty-four different ports; namely, at Brest and Boston of
Ferrel's list, and at all the stations named in Baird and Darwin's list of harmonic

Table XV. — Corrections to the Moon's Mass, deduced from Observations of the Tides.

.

o

u

<u

^2

i-

s

3

X

y.

Name of Station.

1 Penobscot Bay, Me ....

2 Boston, Mass. (336^.196.)

3 Sandy Hook, N. J

4 St. Thomas, West Indus . .

11
12

13

14
IS

16

17
18

19
20

21
22
23
24

25

26
27
28
29
30

31
32
33
34

Falkland Islands, South Atlantic

San Diego, Cal

Fort Point, San Francisco, Cal .

Astoria, Oregon

Port Townsend, Wash

Freemantle, West Australia . .

Port Blair, Andaman Islands
Moulmein, Burmah ....

Amherst, Burmah

Rangoon, Burmah ....
Kidderpore, near Calcutta .

Diamond Harbor, river Hooghly . .
Dublat, Saugor Is., mouth of Hooghly

False Point, Bengal

Vizagapatam

Madras

Negapatam
Paumben
Beypore
Karwar . .
Bombay . .

-
z

.£ «5

3 S

Years.
6

'9
6

3
1

5
3
3
3

1

3

3
3
2

2
2
2
4
3

2

4

5
5
S

Karachi 15

Aden

Port Louis, Mauritius

Brest, France. (336,11.189.). . . .
Helbre Island, mouth of river Dee . .

Liverpool, England

Liverpool, England

Portland Breakwater, England
West Hartlepool, England . .

4
1

19
10

4
3
4
3

i)u'

0000 50

+ 059

+ 321

265

— 000071
+ 094
+ 223
4- 049

+ 3 87

4
4
+

o-ooi 31
084

3 33

4 74
2 95

<\u"

4 000439

+

3 54

+

278

— O-0O2 09
-j- 0 02

— o-ooi 36

+ 382
+ 296
+ 295

506
4 000007

-f 095
4 242

+ 2 18
4- 212

v

,*(V + V)

-f- 0002 67

4

235

+ 3 59
4 o-ooi 24

+ 386

4 1 14
+ 169
+ 462

4- o-ooo 99
1 77
+ 489
+ 070
+ 129

-)- o-ooi 94
026

4- 207

4 0001 26 4 o-ooi 01

o 34 - 05S

4 006

— o-ooi 03
4 238
4 260

4 1 72

— 060

— o-ooo 62
4 090
4 287

235 — 200 . — 1 56

4 256
107

— 0003 61
370

4- 250
+ 227
4 158

4 0001 21
1 10

+ 4 53

— 2 20

— 0001 08

138

— 2 72
4- 219

+ 034

— 236

— 000005
4 1 11

094
059
3 34

— 0-00052

3 41

490

0000 42
o 16
2 17

> 99 4 > 45

1 08 — 1 72

4 000073

4 131

183
4- 076

— o-ooi 83

— I 29

4 078
+ 084

098 i —

088

2 25

4 0001 46
4 ' '9

+ 527

095

4 0-002 77 4 o-ooi 80

4
+

2 40
278
188

4 1 23
+ 043
4- 070

4 000035

— 1 16
018

4- 032
4- 002

4 0-000 84
4 051

- 275
4 204

I >\ HI!', m »LAR PARALL W AND ITS RELATED O iN I \ S

119

tidal constants where the observations extended over a period of two or more yeai
together with some where they were limited to a single year. The corrections given
in the columns headed Sju', d>", 6ju'" have been computed, respectively, by means of
formula' (367), (371), and (374), from Baird and Darwin's harmonic constants;* and
;is the two last-mentioned formula' deal with the same 'data, we have now to consider
which of them should be preferred.

At St. Thomas the K2 and X tides seem unknown, or at least they are not given
in Baird and Darwin's list. Rangoon is situated upon the eastern arm of the Irra-
waddy River, at a distance of 21 miles from the sea; and Kidderpore is a suburb of
Calcutta, on the Hooghly River, 80 miles from the sea, the width of the navigable chan-
nel for 10 miles below that point being only abput 250 yards. These circumstances
render it probable that the values of 6> found from the tides at St. Thomas, Rangoon
and Kidderpore are untrustworthy, and accordingly we shall reject them. An examin-
ation of the remaining data reveals facts which may be tabulated thus :

Column dfi' 6M" 6M"'

Number of plus corrections .... 16 16 24

Number of minus corrections . . 13 13 5

Sum of corrections +488 +824 +3923

If we divide each of these sums by 29, and add the quotients respectively to
0"Oi2 50, the resulting values of the Moon's mass will be 1 : 78*9, 1 : 78*2 and 1 : 72'2.
The latter is so much too large as to place its erroneous character beyond doubt, and
we therefore conclude that formula (371) is decidedly preferable to (374). The
failure of (374) probably indicates that the height of the tides is influenced not so
much by the direct effect of friction upon their amplitudes as by its indirect effect
arising from the changes which it produces in the epochs of certain shallow-water
components, which cannot be separated from the deep-water components with which
they are combined;! and that hypothesis is further supported by the known fact that
when the effects of friction and of the Earth's rotation can be regarded as of the
second order, their effect upon the epochs of the oscillations is also of the second order,
but upon the amplitudes it is of the third order only. J The great difference between
the sums of the corrections in the columns Sju' and Sju" is probably due to the circum-
stance that the diurnal tides are affected by fewer shallow-water components than the
semi-diurnal ones.

The wide range in the values of S/u' and Sju" shows the imperfection of our

present tidal theory, and Professor Ferrel thinks its improvement depends mainly

upon the study of the shallow-water terms. He adds : §

With regard to the determination of the Moon's mass, from the results so far as obtained
the relations of the diurnal tides promise better success in the future than those of the semi-
diurnal tides. The diurnal tides are not affected by so many shallow-water components, and
it is probable that these can be determined from the analysis of the observations, since there
are two comparatively quite large components with periods differing from those of any others, and
hence can be determined by analysis of the observations; and then from the theoretical relations

t336, p- 55- 1336, P- 5'- 2338.P-299-

I 20 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

given in Schedule III the others can be, at least approximately, determined, and the components
of deep-water tides which they affect can be corrected for their effect. The relations of these cor-
rected results, obtained from the analysis of the observations, should then agree with the theo-
retical relatives, and give a correct mass of the Moon.

Instead of attempting- such intricate and uncertain computations, we shall employ
another method which is simpler and not less promising. It is evident from Table XV
that the value of the Moon's mass given by the tides varies irregularly, not only from
point to point along the coast, but also within rivers and large harbors, and even with
respect to the diurnal as distinguished from the semi-diurnal tidal components ; Sju'
and 6fx" having the same sign at 15 places and opposite signs at 16 places. It is
therefore highly probable that the cause of these variations may justly be regarded
as an accidental one, whose effect will vanish from the mean of a series of observa-
tions made at a sufficient number of stations ; and if so, the best mode of treating
the corrections in Table XV will be to take the mean of those given in the fourth and
fifth columns. By that method the result for each station will be h(d/*' + $m"), as
given in the last column of the table ; and then the question of weights arises. At
first sight observations extending over a long period seem likely to be more exact than
those limited to a single year, but in reality there is little difference between them,
because the accidental errors at any station are generally small compared with those
due to constant causes. The arithmetical mean will therefore be the most probable
result, and after rejecting St. Thomas, Rangoon and Kidderpore, we have from the re-
maining 29 stations in the fourth and fifth columns

Sju' = -f-o-ooo 168 ±0000305 ( ~ \

V = + 0-000284 ±0-000348 ^3/5'

Regarded as a single series, the 58 corrections in the fourth and fifth columns give

£(£// -f- Sjli") — + 0000 226 i cooo 230 (376)

while the 31 corrections in the last column give

i(<fy/ -f Sju") z= + o-ooo 214 ±0-000 183 (377)

As (377) contains corrections from Boston and Brest which are not included in
(376), these two results are not strictly comparable, but doubtless the probable error
of the former has been somewhat diminished by the circumstance that it is based upon
the value of \{Sjj! + #//') for each station, instead of upon the entire series of values
of 6ju' and 6/x" . To estimate what probable error (377) would have had in the latter
case, we may imagine the number of corrections in (376) increased from 58 to 62
while their individual accuracy remains unchanged, and thus we shall find

± o-ooo 230 -^(58/62) = ± O'OOO 222 (37$)

In order to ascertain whether any of the corrections are abnormally large
Peirce's criterion was applied, and the limit for the rejection of one observation was
found to be 0006 79 in forming (376), and 0003 70 in forming (377). All the correc-
tions are therefore well within the limit, and none should be rejected. Nevertheless,
it may be thought that with the comparatively small number of observations available.

ON I ill SOLAR PARA1 I A.\ AND [TS R] I ATED < ONS1 wis

121

the final result is injuriously affected by the larger corrections, and t<» test thai theory

all corrections greater than 000200 were arbitrarily rejected from the last column <>t

the table, after which {$77) became

J(tf/< -(- S/u") — — 0000018 ±0000 146

(379)

Very likely (379) is nearer the truth than {377), but as no satisfactory reason can
be assigned for rejecting any of the corrections, we shall adopt for Sju the value (377)
with the probable error {37S) We therefore have as our final result from the tides

<5 M = + 0000 2i4± 0000 222
Mass of Moon z= 00 12 7 1 4 ± 0*000 222 =

(38o)

78 653 ±1374

The mass of the Moon employed in previous sections was derived from the lunar
parallax, and if the probable error of the observed value of that quantity is taken
to be ±0202 74", as given in Table VI, the resulting probable error of the mass is
±0000 180; whence it follows that the mass from the tides has only one-third less
weight than that from the parallax.

27.— A MORE COMPREHENSIVE LEAST SQUARE ADJUSTMENT.

The discussion in Sections 25 and 26 shows that the flattening of the Earth, and
the mass of the Moon from the tides, should be included in the least square adjustment,
and we have now to develop the equations necessary for that purpose.

For the adjustment of the flattening a symbolic correction to its observed value
must be introduced in the conditional equations, and we shall do that by the indirect
method employed on page 1 1 1, because any other process would entail needless compli-
cations. The requisite numerical data corresponding to the two values of e used on page
1 1 1 are given in Table XVI ; those for e' = 1/293-5 having been collected from pages

Table XVI. — (Quantities employed in forming the Conditional Equations (382).

<

<•*

a

'3S-3

oV%

Log. coefficient of 1st equation . .

2<i equation . .

3d equation . .

4th equation . .

5th equation . .

r

Coefficients of 6th equation . . . -j

i
I

Log. coefficient of 7th equation . .

I : 293466 3
0-006 S03 4S 1 o
20 926 202 feet.
3256 872 feet.
0997 742 4^2
0-002 310 461 5

278499322
5303 I24S — 10

4681 9624 — 10
8-9124816-0
7-5260362-0
216 236648

375744492
807 952-643

4-665 0707-1 — 10

I : 300-000 O

0006 655 555 6
20 925 320 feet.
3-256 S3 1 feet.
o-997 791 32

O-O02 3IO336 I

2-78498 1 ss

5-3031248 — 10

4681 9624 — 10

8-9124633-1

7-5260178-9
21625360

37574466
80801598
4-665 03665 — K

122 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

57, 61, 63, 105, and 106, and those for e" = 1/300 having been specially computed. The
coefficients in the second column of the table agree with those of the equations (185),
both having been computed with the value e' of the flattening ; but the coefficients
in the third column do not agree with those of the equations (338), although both
were computed with the value e" of the flattening. The difference arises from the
circumstance that in (338) the equatorial semi-diameter of the Earth is regarded
as independent of the flattening, while in Table XVI that assumption is abandoned
and a has been found from (356), withcterr 1/300— 1/293-466 3 = — 0000074 213.
Then e2, p2 , and l35.3 were found from (326), (7), and (323), and the remaining quantities
from (152), (153), (162), (181), and (183).

For the direct adjustment of the Moon's mass it is only necessary to modify the
numerical coefficient of (157), and to include the corrected expression among the con-
ditional equations. It is the last of the group (382).

If now, in accordance with (361), we put

de=e— 1/293-4663 = £ — 0-003 407 546 (380

and let x represent any one of the coefficients in Table XVI, then the numerical values
given for that coefficient in the second and third columns of the table will be respect-
ively x' and x", and their substitution in (363) will give the general expression for x
in terms of de. In that way the following set of conditional equations for the more
comprehensive least square adjustment was obtained :

ooy vx-p — [2 784 993 22 +0*152 Sde]( t , M j

1 -I- M
oor v2=p — [5303 1248—10] PQ _j|

o or v3 zzp — [4-681 962 4— 10] PL 71 —

[8 912 481 60 + 0 2464^]
oorv4=jp — 1-2 — 5 _t: j

[7-526 036 20 + 0-246 yde]
oorv5=p-y-> ^T? ±-L-^

., M ( 1 — (216 236-65 — 228 40ode) sin3 P )

o or v6 = jW — 2? < j 77J 7 5 c. j n — — TTt I

( 3-757 444 9 — o-022 9i«£ — (807 952-64 — 853 360a*) sin-* r >

, Ar [4-66507071+0-4589^—10]

o or v~, = 1 + M — L~ J ' —'- . .. ,7

sin1 P

Proceeding as in Section 2 1 , we have next to differentiate these equations with
respect to all the quantities expressed symbolically, and in doing so it must be remem-
bered that (204) and (70) require that when E becomes E + dE, §£ shall become
|> — 31 ji6"dE, and when e becomes £ + de, P shall become P+5062"^. The re-
sults are

ON rill SOLAR PARALLAX AND ITS RELATED CONSTANTS. I2t

o - r, + dp + 3 ( , ^j_ M j dM - -4g d E - 0-352 5p . <fc

o=r2 + <(p __^M^/M-{;(,/P + 5o62"^)-(^/Q

o = va + dp +ff:^1p«-f (dP+5o62»'«ie)-£dL

o = v4 + tfp +^V +^</0- 0-5678^.^ g

ori/'s + rf]) + |?dV+^<Za — 0-567 8/>.fte

o = V6 + rfi-|(^-3i 716" dE) + [2-9133]*

1 '//sa r ,i«, 8in2PcOsP.rfP

+ 3 arc 1 (# — |o"S72 46 1 il ) r — ?— -. r , . , ,,

10 v* l 0/ t j &j [5-23996— 10] — [0-572 46J sin3 P

o = v7+ cCSl + 3 arc 1" cot P ( 1 + M) dP

+ ( 3 arc 1" cot P 5 062" — 1-056 8)(i + M) de

To secure the utmost accuracy in the subsequent computations, the coefficients in
(383) should be reduced to numbers with values of the quantities involved midway
between the observed and adjusted ones. The following- were used, and they fulfill
the required condition quite approximately :

1

p — 8809" %\= 9224" a — 20460" V = 186342

P —342267 Q= 12520 (9=497 58 M — 0012520

g = 50"358 L= 6518 £=0-000003030750

Their substitution in (383), and the addition of the appropriate v to each equa-
tion, gave rise to the following expressions :

ozz vx + dp + [0-4624]^^! — [5-9863] c/E — [04921]^

o = v2-\-dp — [i2459](/M — [74106— 10] dP— [88473 — io]r/Q — [11 149]^

0 = vz-\-dp + [2'84i9]c/M — [74106— io]dP — [o i3o8]^L — [11 149] de

o — i\ + dp +[56746— io](Ar+ [8-2481 — io]dO — [0-699 1 ]</e (384)

o = Vs + dp + [5 6746 — 10] dV -f- [96340 — 10] da — [0-699 1 ] de

o=v6 + m -[92628- io]df + [9-3169— io]<*P+[3764i]dE + [2 9133]*

0 = v74-dM 4- [6948 1 — io]dP4- [0-5343]^

The equations (384) were employed to eliminate seven of the twelve unknowns
which they contain, and in (385) each of the twelve is expressed in terms of the re-
maining five, namely, in terms of dp, dP, t/g, da, de, and known quantities. Adopt-
ing the system of weights given by (188) for the probable errors in (392), and
imagining the symbols in the first column of (385) to be replaced by zeros, the
weighted normal equations (386) were formed in the usual way : but in them the
differentials are inclosed in parentheses to indicate that they have now become correc-
tions to the observed quantities. The solution of the normals gave the values of (^p),
(dP), (dg), (da), and (de) which are contained in (3S7), and by substituting these
values in (385) all the other corrections in (387) were obtained.

I 24

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

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o

ro

o

in

r^-

in

fn

vC

r^

t^

PO

■*

N

l^

O

PI

*C

ro

t»J

■+

r^

t^

oo

O

o

X

rn

O

_

in

O

oo

PI

■*

■■

■*

00

oc

o

O

M

."-

t~~

o

o

PO

in

oo

"-■

in

c*

O

o

NO

M

8

o

On

M

NO

NO

O

fo

■*

o

o

NO
O

rn
O

8

o

ON

o

ro

rn

r^

NO

o
o

8

6

b

b

b

b

b

b

fn

ON

b

b

1

i

+

1

+

+

+ + +

1

II

II

II

II

II

II

II

II

II

II

§

cT

86 1$

C?

J*

M

^

>

ST

S

^^

\i

■y

*s

"e

"«

"§

*

>»

•«

->2

Ss

1.26

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

As a preliminary step toward finding the probable errors of the adjusted quan-
tities, we substitute in (387) the values of vu v2l v3, etc., from (384), and thus obtain
the equations (388), which are to be read in the same way as those on page 67.
Then, by the process described on page 71, and with the probable errors of the
observed quantities from (392), the results given in Table XVII were deduced.

Reverting now to the flattening of the Earth, there can be little doubt as to what
the results are from the geodetic arcs, precession and nutation, and perturbations of
the Moon; and for the pendulum experiments Helmert's reduction seems best, partly
because it is based upon more data than any other, and partly in consequence of the
discussion on page no. Accordingly, the following observed values of f, from pages
5, 97, 95, and 102, are the only ones to be considered:

Method of determination.

Geodetic arcs, Clarke . .
Pendulum experiments, Helmert
Precession and nutation . . .
Perturbations of the Moon

1 1293-47 = 0-003 407 503
I : 299-26 = 0-003 341 576
I : 297-67 = 0-003359425
I : 294-93=0-003390635

Equations No (388).

0

Logarithmic coefficients for compating-

-

{dp)

(,/P)

(4)

{dm

{dQ)

(,/L)

(#)

+ 9892 8 -

i°

— 8-469 4 —

10

— 6-667 4 — IO

-S-7'3 9 - '0

— 0498 5

-87195-

10

(,/P)

— 6-168 1 -

10

+ 9662 5 -

10

— 5540 3— 10

+ 7586 3 -10

-81348-

10

+ 8-1365-

10

(rff)

-77470-

10

— 8-9180 -

10

+ 70634— 10

— 9-1100— 10

— 9-096 9 —

10

+ 9-055 0 -

10

(<®)

+ 8-484 2 -

10

+ 9655 2 -

• 10

— 7800 6 — 10

+ 9-847 2—10

+ 9-834 1 -

10

— 9792 3 -

10

(*Q)

— 7-2790 —

• 10

— 7-212 4 —

10

— 4-7978 — 10

+' 6-844 3 — 1°

+ 9-987 8 -

10

+ 5-987 3 -

10

(dL)

— 81800-

10

+ 9-893 7 -

10

+ 7435 8—10

— 9-4824 — 10

+ 8-667 6 -

10

+ 9-5100-

■ 10

(da)

+ 9-435 2 --

- 10

+ 9299 8 -

- 10

+ 6-745 2 — 10

— 8791 8 — 10

+ o-593 8

+ 8-839 6 -

10

(dd)

+ 6-886 7 -

- 10

+ 67513-

10

+ 41967—10

— 62433— IO

+ 8-045 2 -

10

+ 6-291 0 —

10

(,/V)

+ 5-504 7 -

10

+ 5-369 3 -

- 10

+ 28147 — 10

— 4-861 3 — 10

+ 6-663 3 -

10

+ 4-909 1 -

10

(dE)

— 4-809 6

— 4-447 i

— 20713

+ 4-1180

— 5967 4

— 4298 8

(,/M)

+ 95oi 1

- 10

+ 0898 2

+ 84529— 10

— 0-499 6

+ 0-8584

— 0-825 x

(de)

— 05678

+ 3-2506

— 9-406 8 — 10

+ 1-453 1

— 1-6844

+ 1-6113

Factors.

Logarithmic coefficient.-, for computing-

(da)

(dd)

(dV)

(dE)

(dU)

(A)

{dp)

+ 9'649 3 -

• 10

+ 10352

+ 27469

— 33522— 10

+ 6-3258-

10

-5734 2-

10

(,/P)

+ 7-2172-

- 10

+ 8603 1 -

10

+ 0-3165

— 0-693 3—io

+ 5436 7-

- 10

+ 6-1209 -

- 10

(<'?)

+ 80389-

- 10

+ 9-424 8 -

- 10

+ 1-1382

— 1-6935 — 10

+ 6-357 3 -

- 10

- 5-653 7 -

10

m

— 8-776 2 —

- 10

— 0-162 1

-1-875 5

+ 2-430 8 — 10

— 7-094 5 -

10

+ 6-390 9 -

10

W)

+ 7-588 3-

- 10

+ 8-974 2 -

- 10

+ 0-687 6

— 1-2904 — 10

+ 4-464 0 -

- 10

— 3631 1 -

10

(dL)

+ 8-5141-

- 10

+ 9-9000 —

- 10

+ I-6I3 3

— 2-301 7 — 10

— 7-1098 —

- 10

+ 6238 6 -

10

(*)

+ 9-5I95-

- 10

— 1-211 5

+ 2-5248

+ 3-4476— 10

— 6-409 0 —

- 10

+ 5365 9-

10

(dd)

— 7-2770-

• 10

+ 9'979 5 -

- 10

+ 9976 5 — 10

+ 0-899 l — IO

— 3860 4 -

10

+ 2817 1 —

10

(dV)

+ 5-495 0 -

- 10

+ 6-88o 9 -

- 10

— 8-594 3— 10

+ 9-517 1 — 20

— 2-478 5 -

- 10

+ 1-435 4-

- 10

(dE)

+ 5-1192

+ 6-505 i

+ 8-223 9

+ 99702 — 10

+ I-73I4

— 0928 6

(dU)

— 9798 1 -

- 10

— 1-1840

— 2-900 8

+ 3-45o 1 — 10

+ 9994 2 -

- 10

+ 72689-

10

(de)

+ 0-4125

+ 17984

+ 35125 -43034—10

+ 8-9156-

10

+ 97" 1-

10

ON THE SOLAR PARALLAX AND ITS RELATED CONSTAN1

127

TABLE XVIL — Computation of the Constants rrquiri <l for Jin'liny the Probable Errors of tin A/ljusi, ,1

Quantities.

utity.

P

V

s

h\

73 45'

I.
203

^ /

35 3«3

1

2

1 981

§* 1'

3

12 737

2

2 179

272

2 745

7 2

2

0

61370

1 021

10

79

| P

1 166

256

50I

1 10 36

5 843

48195

5 Q

4428

3

'I

598

9 354

1

2 l

586

1569

I90

23605

55

117 112

V)

89784

48

4

4639

18638

578

£ »

6175

3

0

319

12812

40

■8 v

i3244

7

1

684

27492

85

S3 E

10725

2

0

444

22 182

102

| M

5°

3i

4

4918

257

^203

t

Sum
Log. sum

148

34 451

7

8 754

254

1 814

161 692

79 108

614471

371947

338 362

173 157

5-20869

7898 22

4788 50

5-57048

752938

623844

Log. R.

7-604 J4

8949 1 1

7-394 25

7-785 ^4

8764 69

8-119 22

Ra

-(-0-004 021"

± 0088 943"

± 0-002 479"

± 0006 099"

± 0058 169"

±0013 159"

Quantity.

a

6

V

E

M

c

. /

147 IO

86936

23-06

37 446

332

22

8* r

398

235

006

36

1094

25 551

r i

7

4

o-oo

1

32

1

a »

4 475

2646

071

912

19382

759

2 Q

18 399

10879

291

4667

104

2

1 1-

2 731

1 614

o-43

1 027

42442

768

5 «

52967

32048

i3'56

95 061

796

7

JS "

37256

22029

o'93

6 537

55

0

■8 V

12665

7489

1 196-2

14 021

117

1

44627

26388

7-23

"345

75

2

= M

194

"5

003

39

882

'7

e
Sum
Log. sum

73

43

o-oi

44

736

25645

797 595

478858

1 24513

171 136

66047

52 775

5901 78

8- 680 20

3-095 20

3-233 34

0819 85

072243

Log.R.

7-95089

9- 340 10

1-54760

1-61667

5 409 93

5361 22

R.

±0008 93 1 "

±0-21882"

±35286

±4 1369

± 25 700

± 22 973

For use in the least square adjustment we adopt the arithmetical mean of these
results, with an assumed probable error equal to half the difference between the
greatest and least, namely:

e zz 0-003 3 74 785 ± 0000 03 2 964
= I : 296315 2± 2-8946

(389)

I 28 r>N THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

«

The substitution of (389) in (381) gives

(h = — 0-000032 761 (390)

whence, by (70) and (75)

P = 3 423-08" + 5 062" dem 3 422-914 16" ±0-121' (39 1)

The values of the observed quantities employed in the more comprenensive least
square adjustment are those given in (143), (391), (89), (90), (66), (67), (91), (92),
(101), (132), (380), and (389), all of which are collected in (392) for convenience of
reference.

p = 8-834" ±00086"

P ==3422-914 164" + 5 062" cte ±0121"

g= 503586" — 31 71 6"rfE± 0-00248"

$= 9-233 i"±o-oii 2"

Q= 12546" ±035"

L = 6514" ± 0-016"

a — 20-466" ±0011" K^ '

0 = 497'Os ± ro2s

Vz 186347 ± 36 miles

E = 0-000003 005 097 ±0-000000015056

M = 0-012714 ±0000222

e =0003 374 785 ±0000032 964

The adjustment was effected as follows:

1. To allow for the change in the flattening from 1 : 2935 in (382) to 1 : 296 3
in (392), the corrected conditional equations (393) were computed from (382) by sub-
stituting therein the numerical value of de from (390).

2. The residuals, (394), were found by substituting the observed values, (392),
in the corrected conditional equations, (393).

3. The corrections by adjustment, (395), were found by substituting the resid-
uals (394) in the formulae (387).

4. The correctness of all the numerical processes involved in passing from (382)
to (395) was checked by the well-known relation

\jmn] + 100 \_an\(d£) + \bri\{dp) -f cio [cn](c/P)
+ [dri](d^) -f [en)(da) — [pvv] = o

where [paw] is the sum of the weighted squares of the absolute terms in (385) ; [aw],
[few], [en], [dti], [ew] are the absolute terms in (386); and \_pvv\ is the sum of the
weighted squares of the corrections by adjustment in (395). The result of the check
is given in (396).

5. The incompletely adjusted quantities, (397), were found by adding the cor-
rections (395) to the observed quantities (392).

ON THE SOLAR PARALLAX AND ITS RELATED l ONSTANTS. I -'<,

Corrected Conditional Equations.

oorv1=j)- [2784988 2j(t^m)
oorv2=jp —[5*303 1248— io]PQ J — -^
ootvz—P — [4-681 9624— 10] PL - M
oor„,=1,-I»2!^35] (393)

[7^26028 i]

o or y6 = ^ - yg

~ M "J 1 — 216 24413 sin3 P )

oor«6 = #-g J 3757445 7-807980-60 sin8 P )

[466505568-10]
o or «, = 1 + M - 1 __

First set of Residuals.

Vl = + 0-075 184" r5 - + 0-030 149"

v2 = — 0018 582 v6 = + 0079 261 (^

03 = + 0*294 925 r7 = + 0-000662

v4zr + 0-007 374
Firs* Approximation to the Corrections by Adjustment.

(dp) --0-02487956" {da) -- 001161952'

(rfP) =-0-15306762 (d0) = + roo38i4538

(dg _ + o-ooo 103 67 (dV) =-10163 88 miles (395)

(<*§) = - 0-01153896 (dE)= + 0-000000050940

(dQ) =-0-510094 79 (dM) = - 0-000374748

CrfL) = + o-oo7 861 33 («k) =- 000004424841

[pm] +004555243

100 [an] (de) — 5 99Q91
[&n](a» — 2575443

o*io[cn](dP)- 553281

[*•](*» - I21 , M

[en] (da) - 524440 U9°;

Sum +000302867

[^w] +0003028 10

Check 000000057

G9S7-

I ?0 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

First Approximation to the Adjusted Quantities.

p — 8809 120" a = 20-454380"

P = 3422;537 113 Q— 498-004*

i>= 50357088 Vz 18633684 miles

§^= 9221561 E =z 0000003056037

Q= 124-94991 M= 0-01233925

L= 6521861 e — 0003330537

(397)

6. By substituting the value of £ from (397) in (381), the corresponding value
of de was found to be — o 000077 009, and by substituting that in (382) the corrected
conditional equations (398) were obtained.

7. When the quantities (397) were substituted in the conditional equations
(398), they gave rise to a second set of residuals (399), thus showing the adjustment
to be incomplete. (396) proves that no error exists in any of the numerical operations,
and it is easily seen that these residuals arise from the neglect of terms of the second
order in forming the differential equations (383).

8. The better adjusted quantities (400) were obtained by substituting the
residuals (399) in the formulas (387), and adding the corrections so found to the
quantities (397).

9. By substituting the value of £ from (400) in (381), the corresponding value
of de was found to be — 0-000076 492, and by substituting that in (382), the corrected
conditional equations (401) were obtained.

10. A third set of residuals, (402), was found by substituting the quantities
(400) in the conditional equations (401).

Second set of Corrected Conditional Equations.
oorr1 = # — [2784981 45]/ — — V

ooyv2— p — [5303 1248— 10] PQ ! ±—

1 — M

1 AT

oor r3= p — [4-681 962 4 — 10] PL— - -

[8-91246262] , ON

OOXV^—p — L ^—~ J . (393)

[7526 OI 7 20l

o or Vr. = )> — l / j / j

oor*6=tJ-^-_J

, Af [4'66s 035 xi — iol
o or v1 z=. 1 + M — — — - J J J *

sin^P

— 2 16 254-24 sin3 P

4467 — 808 01 8-36 sin3 P

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. I 3 I

Second set of Residuals.

Vi =z -+■ O'ooo 1 40" vs = + o 000 008"

r,- + ooooio7 ',; = + o 000 1 1 8

V3 — + 0"004 322 V7 zr + O'OOO 000 06

v4 — + 0000 030

Second Approximation to the Adjusted Quantities.

p — 8 S09 050"

« zz

20-454 5I3

Pr=3422"542 144

0 =

498-005 98 I8

jg = 50*357 096

v=

186 337-004 miles

|lzr 9220520

E =

0000003 056095

Qzr 124-951244

M =

0-012335 279

Lzr 6522956

£ ZT

0-003331054

Third set of Corrected Conditional Equations.

o or i\-p - [2-784 981 53] ( yXm)

i+M

o or v, —p — [5*303 1 24 8 — 1 o] PQ
oorw3r=^— [4681 962 4— 10] PL

i-M

i+M
M

[7 526017 33]
o or v5 — p — L/ J Tr ' -30-1

X a

«- M\ 1 — 216 254-1 2 sm3P >

oory6 — il — g ^ ryg'

(3757446 7 — 808 017-92 snrP)

, ,f [4665035 61 — 10]

o or V-, — 1 + M — ^ J fJ__.

sin P

(399)

(400)

[8-912462 75]

o or v^ — p — L y TT- ^" , ,.

J V<9 (401)

2%ird set of Residuals.

vx — -\- 0000 00 i" vb— o-ooo 000"

r, = + o-ooo 00 1 v6 = — 0000 o 1 9 (4Q2)

v9 — — o-ooo 040 V-, — — o-ooo 000 034

?>4 zz OOOO OOO

Z%ird A jiprori, nation to the Adjusted Quantities.

}> — 8-809051" a. =20-454 5 I2"

P = 3422-542 157 0 = 498-005 947s

g= 50357096 Vz= 186337-002 miles (4Q3)

ilzz 9-220537 E= OOOOOO3056097

Q- 124951261 Mz= OO12335305

L— 6522940 £ = 0003331057

132 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

11. The finally adjusted quantities, (403), were obtained by substituting the
residuals (402) in the formulae (387), and adding the corrections so found to the
quantities (400).

12. From (403) and (381) the value of de was found to be — 0-000076 489,
and the resulting conditional equations from (382) were sensibly the same as (401).

13. The substitution of the quantities (403) in the conditional equations (401)
gave the final residuals (404), which show that the adjustment is sufficiently complete.

Final Besiduals.

vx ■=. O'ooo 000" v5 rr + o-ooo 000"

v2 == + 0*000 000 v6 — + o-ooo 006 , .

v3 — -\- o-ooo 00 1 v7 = + o-ooo 000 003

Vt rr — O'OOO OOO

The data and results of the computation outlined in (393) to (404) are given iu
Table XVIII, but it yet remains to explain how the probable errors attached to the
various quantities were derived. Putting r" for the probable error assumed for a
quantity of weight unity in equation (188), m for the number of observation equa-
tions in (385), n for the number of unknowns they contain, and \_pvv\ for the sum of
the weighted squares of the corrections by adjustment, we have in accordance with
the procedure described on pages 70 and 7 1

*=^(S) (405)

With r" = O'Oi, m z= 1 2, n — 5, and the values of p and v from (385) and the third
column of Table XVIII, (405) gives q= 1-4091. The probable errors in Table
XVIII result from the multiplication of that value of q into the respective probable
errors in (392) and the respective values of Ra in Table XVII.

The following explanations relate to the quantities appended to Table XVIII :
The masses of Mercury, Venus, and the Earth, together with their probable
errors, were computed by means of formulae (344), (345), and (346), with the value of
dFi rb dFi' given in the fourth column of Table XVIII. The mass of the Moon is
that given in the table, transformed from a decimal to a vulgar fraction.

The lengths of the equatorial and polar semi-diameters of the Earth were found
from (356), with the value of e from the fourth column of Table XVIII, and de from
(381). As the expressions (356) are of the form a = m -f- n.de, if we put r with a sub-
script letter for the probable error of the quantity symbolized by the subscript, we
shall have

ra2 = rm2 + (n>\)2 (406)

ON THE SOLAR PARALLAX AND I l> RELATED CONSTAN'I

133

Table XVIII.— Final Results for the Epoch L850.0.

Quantities.

Observed values.

1 lorrections by ail-
justment.

Adjusted values.

// //

//

// //

P

8-834 4: 0-012 12

0024 95

8809 05 -)- 0005 67

V

3422-69281-!- 0-17050

—

o- 1 50 65

3422542 16 J- 012533

i1

50-356994. 000349

+

o-ooo I I

50357 10 J- 0003 49

%

9-2331 -t 00157s

—

0-012 56

9-220 54 -J- o-ooS 59

'2

1 2546 4- 0493 iS

—

0-508 74

124-951 26 4. 0081 97

L

6-514 -j^ 0-02255

+

o-ooS 94

652294 J- 0-01854

a

20466 J- 0015 50

—

o-oi 1 49

2045451 -|- 0-01258

e

4970s J- 1-437 2S«

+

1-005 95s

498005 95° 4. 0-30834*

V

186347 -J- 50728 miles

—

9-998 miles

186337-00 -J- 49-722 miles

E

o-ooo 003 005 097
_|- o-ooo 000 022 625

}+

51000 |

0-000003 056097
. -J- o-ooo 000 005 829

M

0012 714
J- 0000 312 820

}-

378 695 {

0012335305
-{-0-000036 214

e

0003 374 785
-j- 0000 046 450

}-

43 728 {

0003331057
-j- 0000 032 37 1

Mass of Mercury = g!358 223 ±0-072441 = ^_1_

3000000 83746724=1765762

Mass of Venus = °'^2 ^7 ±°'°°* 5°3 = .

Mass of Earth =

Mass of Moon =

401847 4089684^1874
_ 1-084 720 ± 0002 069 __ 1

354 936

3272144=624

81-0684=0-238
Earth's equatorial semi diameter =

Earth's polar semidiameter
One Earth quadrant

a

20 925 293 ±4094 feet,

— 3 963-1244= 0-078 miles.
= 20S55 5904=325-1 feet,
= 3 94992 2 4-0062 miles.

= 393 775 8r9±4 927 inches,
= 32 814 652 ±4106 feet,

— 6 2 14-896 ±0-078 miles.
1

Earth's flattening- —

a 300-205 4= 2964

Mean distance from Earth to Sun = 92 796 950 4= 59 715 miles.

Mean distance from Earth to Moon = 238 85475 4= 9 916 miles.

Length of seconds pendulum = 3251045+0017356 sin2 cp feet,

= 39'Oi 2 540 + 0-208 268 sin2 cp inches.

Acceleration by gravity, per second of mean time

= 32-086 528 + 0-171 293 sin2 cp feet,
— 97798864-0052 2 10 sin2 9) meters.

'34

ON THE SOLAR RARALLAX AND ITS RELATED CONSTANTS.

Table XVIII gives r£ = db 0*000032 371, and upon the assumption that the
probable error of the length of a well measured geodetic arc is about one part in
150000, (356) gives

rmfora = ± 139-5 feet

rm for b z= ± 1390 feet

With these values the probable errors attached to the adjusted lengths of the Earth's
seniidiameters were found from (406).

A quadrant of the Earth, measured from the north pole along any meridian to the
equator, is the theoretical basis of the metric system, and in computing its length from
our values of the Earth's polar and equatorial seniidiameters the following formula was
employed :

One Earth Quadrant

^(q+.&)<I + i/a-6\2 +

1 /a — b\A 1 ra — b
( \\a-\-bJ ' b^a-\-bJ 256 \a-\-

)\A . 1 /-a — b\6 . ,
>)+27o(r7+-J+etC-

(407)

The distances of the Sun and Moon from the Earth, together with their probable
errors, were computed by means of the formula

D — a cosec^ rb s/[(cosecj> . ra)2 + (D cot^ arc 1" rp)2]

(408)

where a is the Earth's equatorial semidiameter, ra its probable error, D the distance
corresponding to the parallax p, and >-pthe probable error ofjj. The expression for the
probable error of D was found in the usual way, by differentiating a cosec^J, squaring
the several terms, and replacing the differentials by the probable errors.
The equations (321) and (12) give

'=<<--?

45

- esnr q> . TOrt . ,

(409)

from which the length of the seconds pendulum was computed with the adjusted values
of a and e from Table XVIII, ?45from (10) and ^from (17). The expression for the
acceleration by gravity then followed from the usual formula, g — ttH.
Replacing sin2 <p in (409) by its equivalent, |(i — cos 2<p), we get

I=I,S + -}(k - ^ ) -Te - -^ COS 2?

the diffei'ential of which is

<1/ = -\-{ 1 -\ cos 2cp\dli5 — - „ < I 4 — ■ >COS 2ffl

( 2— £ ) t* I 2 — 8)

+ {la — T2(T \2C0S 2<P

or, after substituting the numerical values of Z45, a, e, and tu

dl=z + (1 + o-ooi 668 307 cos 2<p)dl45
— 0000000000675 cos 2 cp. da
+ 1 628 234 cos 2cp . de foot

(410)

(4ii)

(412)

ON THE SOLAR PARALLAX AND ITS RELATED CONSTAN

135

Then writing r, for the probable error of I, r0 for that of /,.,, and substituting the
numerical probable error of e from Table XVI [I, we find with sufficient accuracy

rx = =b V(ro2 + o-ooo 000 002 778 cos2 2cp) foot

(4i3)

The value of r0 seems to be about ± 0-000040 of a foot, but its uncertainty is
so great that no probable error is appended cither to the length of the seconds pendu-
lum or to the expression for the acceleration by gravity.

Formula? (408) and (413) are sufficient to give the probable errors of I) and I
witli all needful exactness, but nevertheless they are only approximate. Rigorous
accuracy would require the differentials of the adjusted quantities to be expressed in
terms of the original observed quantities, and then the formula' would become much
more complicated.

28.— THE ELECTRIC CONSTANT v.

Maxwell has employed the symbol v to denote the ratio of the electrostatic to
the electromagnetic unit of electricity, and has endeavored to prove that v is a velocity
identical with the velocity of light. The correctness of his theory is now generally
admitted, but its experimental verification is still a matter of interest, and for that
reason an attempt has been made to collect all the published values of v in Table XIX.

Tablu X[X.— Values of the Electric Constant v, found by various Experimenters.

Dale.

Authority.

1857
1869
1868

•873

1S79

1879

1880

1882

1883

1884

1888

1889

1889

1889

1890

Kohlrausch & Weber (359, p. 264)* . .
W. Thomson & KING (360 and 362, p. 417)

Maxwell (361, p. 651)

M'Kiciian (362, p. 427)

Ayrton & Perry (363,11. 14O

HOCKIN (364, p. 288)

Shiiia(365,P.436)

Exner (367, p. 1 1 1, and 369, p. 301) . . .

J.J. Thomson (368, p. 721)

Klemen6ic (369, p. 328)

IIIMSTKDT (37O, p. 136)

W. Thomson (372, p. 312)

Rowland (371, p. 296)

Rosa (372, p. 312)

J. J. Thomson & tjEARLE (373, p. 378) • •

Observed

v

3107
2846
288-0
293-2
2980
2988

299-5
301- 1
2963
3019
3009

Corrected
v

2982
300-0
299-6

3107
2808
284- 1
289-3
2960
2968

295-5

292-3
296-3
301-9
3009
3004
298-2
300-0
299-6

*Therof KOHLRATJSCH & WEBER is w/2.

The results obtained by the several experimenters are given in the third column o the
table, but they depend upon various standards of resistance, and must be reduced to
absolute measure before comparison either with each other or with the velocity of
light That reduction has been effected upon the assumption that the true ohm is
0^8664 of the British Association ohm, and the results are given in the last column
of the table. Since 1857 the observed values of v have approximated more and more

I36 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

to the velocity of light, and if we reject the results obtained by Koiilrausch &
Weber, Sir W. Thomson & King, Maxwell, M'Kichan, and Exner, the ten remain-
ing determinations give

^ = (29856^ 155) 106 centimeters per second (414)

while the velocity of light from Table XVIII is

V rr (29 988 ±8) 106 centimeters per second. (41 5)

The difference of these results is only 0-440 of one per centum, and is so much
less than the square root of the sum of the squares of the probable errors as to leave
little doubt that V and v are absolutely identical.

29.— SUPPLEMENTARY DATA.

The following data are inserted here to remedy omissions which have occurred
in the preceding pages from various causes.

In 1863 Archdeacon Pratt proposed a method of deducing the size and figure of
the Earth, which was based upon the idea of eliminating the effects of local attraction
from the amplitudes of the geodetic arcs employed, and by it in 1871 he found from
the Anglo-Gallic, Russian, and Indian arcs*

azz. 20 926 184 feet

■

b — 20855 304 feet
e— 1/295-2

In 1876, from a series of pendulum observations made at JT) stations, whose lati-
tudes ranged between + 79° 50' and — 62 ° 56', Dr. A. Fischer foundf

I r= 099 1 o 1 1 + 0-005 : °5 sm2 <P meter

In 1884, from the series of observations used by Dr. Fischer, Dr. G. W. Hill
found J

/= 0-992 714 8 meter

+ 0-005 089 op-4 (sin2 cp \

+ OOOOO97 9/3_4COS2 <£>COS (2G0' + 29°04')

— o-ooo 135 5 p~5f sin3 <p — 3 sin cp j

■ 1 (4l6)

+ o-ooo 542 1 p_5(sin2 cp— \ cos <pcos (go' + 2170 51')

-foooo 264 op-5 sin cp cos2 cp cos (20/ +4° 49')
-f o-ooo 1 24 8 p~5 cos3 cp cos (30/ + 1 io° 24')

+ o-ooo 1 48 9 p~6 ( sin4 cp sin2 cp -\- ^ )

V 7 35^

*1SK» P- J77- Compare also 25^ and 22^. t24^» co1- 87- tS7'A> P- 339-

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. 137

-+- 0*000 738 6 p ~ c f sin3 cp — ^ sin cp \ cos </> cos (r«/ -}- 30 02')

+ o-ooo 217 5 p~°f sin2<p — ") cos2 <pcos (2m' -f 2620 17') (4 16)

-f- o-ooo 3 1 2 6 p~6 sin <p cos3 <p cos (3ft)' -f- 1480 20')
+ 0*000058 4 p-6 cos4 <pcos (40/ + 2480 19')

where cp is the geocentric latitude, 00' the geographical longitude, and p a factor, vary-
ing with the latitude, such that the radius of the Earth at latitude cp is ap.

In the course of their investigations respecting nutation, Bessel, Peters, and
Nyren have made use of pendulum experiments to find the values of certain func-
tions which they called P and P'. The relation between I, P, and P' is*

I = 1 + P sin2 cp' + F (sin3 cp' - 3 sin cp'\ (417)

where the length of the pendulum at the equator is taken as unity. With that unit,
formula (9) gives

Z=i+(^-*)sin> (418)

and by equating (417) and (418)

(™* _ e\ sin2 cp - P sin2 cp' + P (sin3 cp' -^ sin cp'^j (4 1 9)

At the poles cp — cp', and as all powers of the sine are there unity, (419) reduces to

ezzi^-P^P (420)

hh 5

or, with our adjusted values of a, /0 and tx

£ = 0-00866952 — P=Fo-4P' (421)

in which the double sign is to be taken negative for the northern, and positive for the
southern hemisphere of the Earth. To distinguish the two values of e thus arising,
we shall call the former e\ and the latter e".

In 1818, from pendulum experiments at 31 stations, Bessel foundf

P — + 0-005 444 8 P' — + o-ooo 668 9

whence, by (421)

£' = 1:338-17 e" — 1 1286-34 %(e' + e")= 1:310-11

In 1 84 1, from experiments at 54 stations, Dr. C. A. F. Peters foundj
P = + 0-005 233 F = — 0-000334

* 1, p. 130. f 1, p. 131. Ji09,p. 170.

I38 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

whence

e' — 1 : 280-10 e" zz 1 : 30276 J(e' -f *") r= I : 290*99

In 1872, from experiments' at 74 stations, Nyrkn found*

P = + 0*005 ] 94 1J' = _ o*ooo 1 34

whence

f'zz 1:283*36 £"zz 1 1292-23 £(<?' + e") — 1 : 28773

All these values of the Earth's flattening should have been included in Table XI.

The fifth volume of the "Account of the operations of the great trigonometrical
survey of India" contains a vast mass of data respecting pendulum experiments, but
it was impossible to utilize them in the present investigation, because no general results
are given, either for the length of the seconds pendulum or for the figure of the Earth,
and none can be deduced without a large expenditure of labor.

In 1890 Professor Newcomb rediscussed the observations of the transits of Venus
which occurred in 1761 and 1769, and found from them, for the solar parallax,
879" ±0-034".!

30.— SUMMARY OF RESULTS.

Three essentially different systems of astronomical constants are given in the
preceding pages, namely, (a) in Table VI, a system based upon General Clarke's
spheroid of 1880, whose semiaxes are specified by the equations (3); (b), on page 1 11,
a system based upon Clarke's value of the Earth's equatorial semidiameter, to wit,
20926 202 feet, and adapted to any possible value of the flattening; and (c), in Table
XVIII, a system in which the size and figure of the Earth were included among the
quantities determined by the general adjustment. The latter system is certainly the
most probable of the three, and for convenience of reference the values of all the
quantities involved in it are here collected and appended. They are to be regarded
as the definitive results of the investigations embraced in this paper.

Size, Figure, Density, and Moments of Inertia of the Earth.

Equatorial semidiameter zz a zz 20 925 293 ± 409-4 feet

zz 3 963 124 ±0-078 miles

— 6 377972 ± 124-8 meters.
Polar semidiameter zzfrzz 20 855 590 ± 325-1 feet

zz 3 949-922 ± 0062 miles

— 6 356 727 ±9909 meters.
One Earth quadrant =393 775819=1=4927 inches,

zz 32 814652 ±410-6 feet,
zz 6 2 14-896 ±0078 miles,
zz 10001 816 =1= 1 25-1 meters.

Flattening zz zz — —

a 300205 ± 2-964

2 7 2

Eccentricity zz ---., zz 0006 651018

*i04,p. 57. t 234^2, P- 402.

«>N rtiE SOLAR l'AKAl.l AX AND ITS RELATED CONSTAX 139

Log. p = 9-999 2772 758 + 0-0007245325 COS 2CP

— 00000018 131 cos 4<p
+ o-ooo 0000 047 cos 6cp
q>— q> =688-2242" sin 2<p— 1-1482" sin 4<p + 00026" sin 6q>
Mean density of the Earth = 5-576 +; 0016

Surface density of the Earth =z 2*56 ±016

Moments of inertia of the earth, (C — A) : C = 0-003 2^5 21 = 1: 306-259

C — A = o'ooi 064 767 EVr
A = 13 = 0-325 029 K a
0 = 0-326 094 E'rr

Length of the Seconds Pendulum.

I — 390 1 2 540 + 0-208 268 sin2 finches,
= 3251 045 +0-017 356 sin2 <p feet,
— 0-990 910 + 0-005 29° sm2 <P meter.

Acceleration hy Gravity, per Second of Mean Solar Time.

g — 32*086 528 + 0-171 293 sin2 <pfeet,
= 9-779 886 + 0-052 210 sin2 cp meters.

Length of the Year.

Sidereal year = 365-256 357 S mean solar days,
= 365do6h09m09-3i49
■=31 558 149-314 mean solar seconds

Tropical year = 365*242 199 870d — o-ooo 006 2 1 2 4d{ 5- J

= 365d o5h 48m 46-069* - 0536 75g(*~^5°)

Length of the Month.

Sidereal month =27-^521661 i62d — 0000000 262 4od( \

\ 100 J

= 27do7b43mir5249-o-022 67i8^~^o00j

A r r*(t l800\

Synodical month = 29-530 588 435d — o-ooo 000 306 96° f iqq \

M ». o s/t— iSOO \

= 29d 1 2b 44m 02-84I8 — 0-026 5228t[ 1

Length of the Sidereal Day.
86 164-099 65 mean solar seconds.

Ratio of the Mean Motions of the Sun and Moon.
w = 0-074 Soi 329 112

140

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

Masses of the Planets.

N. B. — The value given for the Earth is the combined mass of the Earth and Moon.

Mercury rr
Venus rr
Earth rr
Mars —
Jupiter rr
Saturn rr
Uranus rr
Neptune rr
Moon =

(8 374 672 ± 1 765 762)
(408 968 ±1 874)
(327214^624)
(3 093 500 ±3 295)
(1 047-55 ±0-20)
(3 5OI'6±o-78)

(22 6oo± 36)

(18 780 ±300)

(8 1 -068^=0-238)

Solar Parallax,
p rr 8-809 05" rt 0-005 67"

Mean distance from Earth to Sun rr 92 796 950 ±59 715 miles,

rr 149 340 870 ± 96 101 kilometers.

Eccentricity of the Earth's Orbit.

ejrrO'016 771 049 — 0*000000424 5 (t— 1850)

— o-ooooooooi 3671 ^— 1

V '°° /

Lunar Inequality of the Earth.
Lrr 6-522 94" ±0-01854"

Lunar Parallax.

Prr3422'542 16" ±0-125 33"
sinP : arc 1" rr 3 422385 1 1" ±0125 t,^"

Mean distance from Earth to Moon rr 60-269 315 ± 0002 502 terrestrial radii,

= 238854-75 ± 9-916 miles,
= 38439601 ± 15-958 kilometers.

Eccentricity and Inclination of the Moon's Orbit.

62 = 0-054899 720
Delaunay's y rr sin \ I rr 0-044 886 793
lrr 50 08' 43-3546"

Mean Motion of the Moon's Node in 365^ Days.

yurr- 190 21' 19-6191"+ 0-141 36"(t~lS°°\

V IO° /

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS. 141

Parallactic Inequality of the Moon.
Q = 1 24-95 l 26" ± °'°8 ' 97"
Secular Part of the Precession.

n — 17-330968" ^=2-183513

^ = (50-357 10" ±0-003 49")^ — 1850) — o-ooo 106 70" {t— 1850)2
go — 230 27' 31-47" ±0345" +0-000006 41" (t— 1850)2
& =(50-236 1 5" ±000349")^— i85o)+o-ooouo23"(^— 1850)2
^ = 23° 27' 3 1-47" ±0-345— 0466 57" (t— 1850) — 0-00000073"^— 1 850)2

Constant of Nutation.

^ = 9-2 20 54" ± 0-008 59" + o-ooo 009 04" (t — 1 850

General Formulae for Precession and Nutation.

[For explanation, see page 86.]
iff =[50 357 IO" +0-000 049 43" (t0 — 1850)] (t — t0)

- o-coo 106 70" it — tQy + V
co =23° 27' 3147" -0-466 57" (^-1850)

— 0-00000073" (/„ — j85o)2+oooooo6 4i"(^ — t0)2 + £L
A =[0-131 84"— o-ooo 18656" (*„— 1850XK* — t0)

-o-ooo 236 53" (* -to)2

m =46-063 08"+ o-ooo 277 25" [(t0— 1 850) + (* — *„)]
n = 20-046 58"— o-ooo 084 94" [(*o — 1 850) + (t — <o)]

ipi =[50-236 15" +0000 22045" {t»— 1850)]^ — ^)

+ o-ooo 1 10 23" {t -t0f + lP
co, =23° 27' 3 1-47" -0-466 57" (*„- i85o)-o-oooooo73/,^0-i85o)2

— [0-46657" + o-ooo 00 1 46" (to— iSso)](t—t0)

— o'ooo 000 72," (t — A,)'2 + ^

tp" = [0-469 51"— O-OOO 006 89" (to — 1 850)] (< — to)

— o-ooo 003 45" (t — to)2
&' - 1 73° 34' 54" + 32-655" Co - 1850) - 8-791" (t-t0)
M = 173° 34' 54" + 32-655" [&- 1850) 4- (<- '«,)]

IF = —

+

+
+

n = +

+

1 7-2463" + 0000 173 2" {t— 1850)] sin a
0-2070" + o-ooo 000 2" (t — 1850)] sin 2 Q
1-2642" + 0000001 2" it— 1850)] sin 2©
02043" + o-ooo 000 2" it — 1 850)] sin 2 £
01 2 73" + o-ooo 000 1" (t— 1850)] sin A0
0-0686" + o-ooo 000 1" (t— 1850)] sin A,!

9-2 205" + o-ooo 009 o" (t — 1 850)] cos go
0-0899" — o-ooo 000 5" (t — 1 850)] cos 2 Q,
0-5486" — o-ooo 002 9" (i — 1 850)] cos 2 0

0-0886" - OOOO OOO 5" (t — I 850)] COS 2 C

142 ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS,

Constant of Aberration,
a— 20-454 5i" ±0-01258"

Light Equation and Velocity of Light.

0 — 498-005 958 ± 0308 34s
V:= 18633700 ±49-722 miles per second,
— 299 87764 ± 80019 kilometers per second.

31.— CONCLUDING REMARKS.

The preceding section contains a thoroughly homogeneous system of constants,
based upon an enormous mass of astronomical, geodetic, gravitational, and tidal observa-
tions which have required more than two hundred years for their accumulation, and
now it only remains to inquire how these constants can be most readily improved in
the future. The first step in that direction will be to examine the sources of their
probable errors, all of which are exhibited in Table XVII, and in doing so it must not
be forgotten that a quantity may contribute little to the probable errors in the table,
either because its own probable error is small, or because it is very large. In the
former case the uncertainty which it introduces will be intrinsically small, while in the
latter it will be rendered small by its small coefficient, and these two cases must be
carefully distinguished.

Table XVII shows that the probable error of the Earth's flattening arises quite
equably from the probable errors of the observed values of P and £, and these are
almost its onlv sources.

Respecting P, we remark that as yet its value depends mainly upon meridian
observations made at Greenwich and the Cape of Good Hope, but a value determined
from observations made at some station near the equator is urgently needed to aid in
ascertaining the true flattening of the Earth. Such a value could be obtained in a few
months by employing the method of vertical transits to compare the position of the
Moon when rising and setting with that of a neighboring star, and the work might
well be undertaken by a small astronomical expedition. If, in addition to that, the
observatories at the Cape of Good Hope, Melbourne, Sydney, Adelaide, Cordoba, and
Santiago de Chile would co-operate with those of the northern hemisphere in system-
atically making meridian observations of the Moon for two or three years, the result-
ing data would doubtless lead to a material improvement of our knowledge concerning
the lunar parallax and the flattening of the Earth. A probable error of ±0-026 89"
in each of the two values of P, found respectively for the equator and for the pole,
would give rise to an uncertainty of one unit in the reciprocal of e.

On page 126 the observed value of e was found from the four values given
respectively by the geodetic arcs, Helmeet's reduction of the pendulum experiments,
precession and nutation, and perturbations of the Moon; but now we must also take
account of the last general adjustment. With the values of ^ and $ from Table
XVIII, formula (288) gives (C — A)/ C = 0003 265 29, whence, by Table IX,
£= 1 : 297-9 1. Further, although in Table XII Hansen's values of the Moon's per-

ON TI1K SOLAR PARALLAX AM) ITS RELATED CONSTANTS.

'43

turbations give £r= i : 294-93, Airy's value, founded upon much better known data,
gives £= 1 :3ii'4.o, and tliere is no way of determining what tin; true; value is. In
short, the general adjustment, the pendulum experiments, and precession and nutation
give a flattening differing little from 1 : 300, the result from lunar perturbations is
uncertain within rather wide limits, and the geodetic arcs give 1 : 293"5. Thus it
appears that the geodetic value stands quite alone, and as it is almost certainly erro-
neous, the probable error of the observed value of e could be largely diminished by
making s depend solely upon the results from pendulum experiments and precession
and nutation. At the present time more pendulum experiments, both in the neighbor-
hood of the equator, and as near as possible to the poles, are among the things most
needed to perfect our knowledge of the figure of the Earth. Experiments in middle
latitudes will contribute little to that end, however useful they may be for other
purposes.

Unfortunately no value of the Earth's flattening can be deduced, either from the
pendulum experiments or from precession and nutation, without making some assump-
tion respecting the unknown constitution of the interior of the Earth. Notwithstand-
ing the difficulties which arise in connection with the rigidity exhibited by the Earth
under the action of the forces which generate precession, nutation, and the tides, the
theory of a comparatively thin crust resting in approximate hydrostatic equilibrium
upon a denser substratum is favored by enough facts to render it very plausible. If
adopted, it necessarily entails the use of the condensation method for the reduction of
the pendulum experiments, but it is not clear that He^mert's numerical result depends
essentially upon that method, because Unferdinger got nearly the same result by
means of a totally different theory. Furthermore, if the interior of the Earth ever
was liquid, Le Gendre's law of the distribution of density therein must be at least
approximately true, and some evidence to that effect is afforded by the substantial
agreement between the values of e found respectively from the general adjustment,
the pendulum experiments, and precession and nutation.

As stated on page 5, the geodetic arcs used in determining the figure of the Earth
are located in England, France, Russia, India, South Africa, and Peru. Hitherto no
great arcs except those of the Lake Survey* have been measured in the United States,
but several have been projected by the U. S. Coast and Geodetic Survey, and some of
these are so far advanced that their completion may be expected in the near future.
All who comprehend the importance of the subject hope that the United States will
not long remain behind other great nations in contributing her share toward deter-
mining the size and figure of the globe we inhabit.

The extent to which the flattening of the Earth deduced from the geodetic arcs
differs from that found by other trustworthy methods seems to indicate that the exist-
ing arcs are not well adapted to determine the relative lengths of the polar and equa-
torial semidiameters. Possibly the difficulty may arise from the preponderating
influence which the ten southern degrees of the Indian arc now have in determining
the length of the polar semidiameter,t and if so, a meridianal arc situated within the

•The Lake Survey arcs were not published by Gen. CoMSTOi k until two years after Clarke's spheroid of 1SS0. The
arc of parallel is in latitude 420, and therefore very favorably situated for determining f.
fi3, pp. 310-31 1.

144

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

United States, and having its center about latitude 350, would be of great value. To
determine the reciprocal of £ with a probable error not exceeding one unit, the
probable error of each of the semidiameters must be reduced to ± 164-4 feet. With
our present resources it will probably be best to find the value of e from pendulum
experiments, precession, nutation, and perturbations of the Moon, and after the ratio
of the Earth's semidiameters has thus become known, their absolute lengths can be
got from the arcs with great exactness.

Table XX has been compiled from Table XVII in order to show as clearly as
possible the relative importance of each quantity in determining the probable error of
every one of the twelve quantities included in the general adjustment. The several
quantities are entered symbolically in the first column of the table, and on the same
line with each of them the principal sources of its probable error are indicated in the
order of their importance by the numerals 1, 2, 3, etc. For example, in determining
the probable error of Q, the parallactic inequality of the Moon's motion, the quanti-
ties whose observed values have 'the most influence are, 1, the constant of aberration;
2, the solar parallax; 3, the velocity of light; 4, the mass of the Earth; and, 5, the light
equation. A glance at the table shows that the numeral 1 occurs once in each of the
columns P, j§, V, and e, three times in the column L, and five times in the column a.
It is therefore evident that in order to improve the system of constants determined in

Table XX. — Origin of the Probable Errors of the Adjusted Quantities.

in

—

3

Principal sources of the Probable Errors.

P

P

1

1

Q

L

a

e

v

E

M

1
2

1

P
P

B

Q

L
a
0
V
E
M
e

2

2

2
2

2

1

3

3

2

2
2

2

4

• •

3

1
1

1
3

I

1

1
1

I

5

4
4

3
3

1
3

4

4

3

3

4

4

the present paper the quantities whose observed values we should endeavor to improve
are, 1, the constant of aberration; 2, the lunar inequality of the Earth's motion; 3,
the solar parallax: and, 4, the constant of nutation. The constant of precession and
the velocity of light are so well determined as to be virtually independent of all other
quantities, while the parallactic inequality of the Moon and the mass of the Moon
from the tides are so uncertain that they have little influence upon anything.

On account of the great importance of the constant of aberration in our adjust-
ment, it seems desirable to recall the fact that at present we have no satisfactory

ON TH1 SOLA] PARALLAX AND ITS RELATED CONSTANTS | j ^

theory of aberration. The constant in question has long been regarded ae the ratio
of the Earth's mean orbital velocity to the velocity of light, but notwithstanding the
plausibility of that assumption it has never been rigorously deduced from the undu-
latory theory of light.

The desiderata for the improvement of the system of constants discussed in the
present paper may now be recapitulated as follows:

i. The parallax of the Moon should be determined by the diurnal method, at one
or more stations as near as possible to the equator.

2. The observatories in the northern and southern hemispheres should co-operate
with each other for two or three years in systematically making meridian observations
of the Moon to improve our knowledge of its parallax

3. Pendulum experiments should be made at a large number of stations, located
partly in the neighborhood of the equator and partly as near as possible to the poles.
Experiments in middle latitudes are also desirable, but somewhat less necessary.

4. New determinations of t ;e constants of aberration and nutation should be
made by as many different methods as possible.

5. The meridian observations of the Sun, accumulated at the Greenwich and
Washington Observatories during the last fifty years, should be discussed in such a
way as to deduce from them the most probable coefficient of the lunar inequality of
the Earth's motion.

6. New determinations of the solar parallax should be made by observing Mars
during its opposition in 1892, and also by observing- such asteroids as may come into
favorable positions for that purpose.

7. The measurement of some of the great 'arcs included in the scheme of the U.
S. Coast and Geodetic Survey should be completed as soon as possible.

There can be no doubt that the observations specified in paragraphs 1, 2, and 3
would materially improve our knowledge, both of the lunar parallax and of the
Earth's flattening; but the probable errors of the constant of aberration, the constant
of nutation, the coefficient of the lunar inequality of the Earth's motion, and the solar
parallax are already so small that it will be exceedingly difficult to reduce them any
further. Nevertheless, the attempt should be made.

The values given on page 139 for the Earth's moments of inertia have been
derived from the adjusted values of i', ill, /i, I, and £ by means of the formula' (288)
and (306). These moments of inertia determine the period of the Eulerian nutation :
and if there is any want of coincidence between the Earth's axis of rotation and its
principal axis of inertia, they also determine the period of the resulting variations of
latitude. Although arising from widely different causes, these two periods depend
upon the same function of the moments of inertia, and are therefore alike, the length
of each being A/(C — A) sidereal days,* or

X ^ 5 25 4 _ 0-997 270 ^ r mean solar days

C-A ~ 366-2564 -"**' -'~C-A
With our values of A and C, the numerical results are 305-26 sidereal, or 304-42 mean
solar days.

6787 10

* 109/,, p. 126, 105;,, p. 569, and 104, p. 18.

146 ON I III. SOLAR PARALLAX AND ITS RELATED CONSTANTS.

The computations in the preceding pages were all made by the author, and as it
proved impracticable to have them checked by any other person, certain parts of the
work were done in duplicate by means of two different sets of formula', and special
care was taken to apply check equations wherever possible. Most of the numerical
operations were effected upon a Thomas arithmometer capable of dealing with factors
having ten or eleven significant figures, and advantage was taken of that circumstance
to increase the accuracy of the checks by carrying the results somewhat beyond the
usual number of decimals. The accompanying probable errors will suffice to prevent
these decimals from creating any false impression respecting the degree of accuracy
attained.

In conclusion the author desires to thank his colleagues, Professors Asaph Hall
and Edgar Frishy, U S. Navy, and his friend, Mr. R S. Woodward, U. 8. Coast and
Geodetic Survey, for the interest they have taken in the preparation of the present
paper, for their kindness in examining much of it, both in manuscript and in proof,
and for many valuable criticisms and suggestions.

32.— BIBLIOGRAPHY.

The titles of most of the works consulted in the preparation of the present paper
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Month. Not. for Monthly Notices of the Royal Astronomical Society (London),
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Phil. Trans, for Philosophical Transactions of the Royal Society of London.

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ON THE SOLAR PARAL1 W AND ITS RE] \ I I D <

I49

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*50

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61. Newcomb (Simon). Investigation of corrections

to Hansen's tables of the moon : with tables for
their application. By Simon Newcomb * * *
Forming Part III of Papers published by the Com-
mission on the Transit of Venus. Washington,
1876. 4to, 51 pp.

62. Newcomb (Simon). A transformation of Hansen's

lunar theory compared with the theory of De-
launay.

Astronomical papers prepared for the use of the
American Ephemeris and Nautical Almanac, vol.
1, pp. 57-107. (Washington, 1882. 4to).

63. Plana (Jean). The'orie du mouvement de la lune.

Par Jean Plana, astronome royal et directeur de
l'observatoire * * * Turin, 1832. 3 vols., 4to.
T. 1, pp. xxiv+794; t. 2, pp. viij-f865: t. 3, pp.
vj+856.
ti4. Stone (E. J.). Note on the coefficient of the paral-
lactic equation in the lunar theory.
Month. Not., 1863, vol. 23, p. 210.

65. Stone (E. J.). A determination of the coefficient of

the parallactic inequality, and a deduction of the
value of the sun's mean horizontal equatorial par-
allax from the Greenwich lunar observations,
1848-1866. (Abstract.)

Monti). Not., 1867, vol. 27, p. 271. .See also vol.
41, pp. 24-33.

66. Stone (E. J). On the determination of the coeffi-

cient of the parallactic inequality in the expres-
sion for the moon's longitude.

Month. Not., 1880. vol. 41, pp. 24-33.

67. Stone (E. J.). Note on some points connected with

the determination of the coefficient of the paral-
lactic inequality.

Month. Not,, 1881, vol. 41, pp. 381-384.

68. Stone <E. J). On some systematic errors in the de-

termination of the semidiameter of the moon from
the Greenwich observations, 1750-1840.
Month. Not,, 1881, vol. 42, pp. 51-54.

69. Stone (E. J. ). Note on Messrs. Campbell and Nel-

son's paper on the parallactic inequality, in the
Supplementary Number of the Monthly Notices.
Month. Not,, 1881, vol. 42, pp. OJ-66.

70. Stone (E. J ). Note on Mr. Neisou's paper on the

corrections to Adam's semidiameter of the moon.
Month. Not., 1882, vol. 42, pp. 301-304.

71. Stone (E. J.). Curves showing the changes in the

adopted diameter of the moon as giveu by the
observations in the Greenwich Lunar Reductions,
1750-1830.
Month. Not., 1882, vol. 42, pp. 374-378. 1 plate.

LUNAR PARALLAX.

72. Breen (Hugh). On the constant of the moon's hori-

zontal equatorial parallax, as deduced from obser-
vations made at the Cambridge Observatory and
the Royal Observatories of Greenwich, Edinburgh,
and the, Cape of Good Hope.

Mem. Roy. Ast, Soc, 1863, vol. 32. pp. 115-178.

ON Mil SOLAR PARALLAX AND ITS RELATED CONSTAN

151

73. Henderson (Thos.). The constant quantity of

the moon's equatorial horizontal parallax, de-
duced from observations made at Greenwich.
Cambridge, and the Cape of Good Hope 111 1832
and 1833.
Mem. Roy. Ast. Soc, L837, vol. 10, pp. '283-294.

74. 01ufsen(C. F. R.). Untersuchungen iiberden Werth

der Mondsparallaxe, die ana den in derMittedes
vori gen Jahrhundertsanges tell ten correspondiren-
deu Beobachtungen abgeleitet werden kann.
Ast. Naoh., 1-37, Bd. 14. s. 209-226.

75. Stone (E. J.). Constant of lunar parallax.

Mem. Roy. Ast. Soc. 1865, vol. 34. pp. 11-16.

PREOESSIOX OF THE EQCIXOXES.

76.

Bessel (F. W.). Ueber die Abanderung, welche die
Bestimmung der Vorriickung der Nachtgleichen,
durch die Konigsberger Fuudamental-Cataloge
erhalt.

Ast. Nach., 1826, Bd. 4, S. 401-406.
77 Dreyer (J. L. B.). A new determination of the con-
stant of precession.

Copernicus (Dublin), 188*2, vol. 2, pp. 1:15-155.
For a review of this paper by E. Schiiufeld, see
Vierteljahrsschrift der Ast. Gesell., 1882, 17 Jahr-
gang, S. 249-259.

78. Hansen (P. A.). Ueber die Reduction der Oerter

der beweglichen Gestirne anf die gleichzeitijje
Ecliptik oder den gleichzeitigen Aequator und das
dazu gehorige Aequinox.
Ast. Nach., 1852, Bd. 35, S. 97-156.

79. Lehmann (W.). Elements for precession, etc., in

bis paper entitled "Die Elemente der Batmen
der acht Hauptplaneten fiir die Fnndamental-
Epoche 1800, Jan. 1, nebst ihren differentiellen
Sacular-Aeuderungen erster und zweiter Ord-
nnng."
Ast. Nach., 1866, Bd. 68, S. 171-172.
Le Verrier (U. J.). Elements for precession, etc.
Annales de l'Observatoire Imperial de Paris,
1856, tome 2, p. 174 : 1858, tome 4, p. 49.
Newcomb (Simon). Reduction of the constants
of precession found by Bessel, Struve, and Nyreh
to a common eqninox.

Vierteljahrsschrift der Astronomischen Gesell-
schaft, 13 Jahrgang, 1878, S. 107-110.

82. Nyreh (M.). Determination du coefficient constant

de la precession an moyen dV'toilesde faible eclat.
Bulletin de 1'Acad. Imp. des Sci. de St.-Pdters-
bourg, 1870, t. 14, cols. 542-574 : Melanges Math.
et Astron., 1867-1872, t. 4, pp. 590-632.

83. Serret (J. A.). Tbe"orie du mouvement de la terre

antour de son centre de gravite.

Annales de l'Observatoire Imperial de Paris,
1859, t. 5. Memoiree, pp. 239-336.
B4. Stockwell (John N. ). [On the precession of the
iquinoxes and the obliquity of the ecliptic] in pp.
171-176 of his paper entitled "Memoir on the
secular variations of the elements of the orbits of
the eight principal planets, (May,

1872.)

Smithsonian Contributions to Knowledge, 1873,
vol. 18, pp. xviij+199.

B0

-1

Struve (O.). Bestiinmnng der Conatante <lir Pi 1
cession, mit Berttcknichtigung der eigenen Bew<

gunn des .Soniiensysti-iiis.

Mem. de l'Acad. Imp. dee Science* de St.-P6tora-
bonrg, Soi. Math, et Phys., 1844, t. 3, pp. 17-124.

86. Struve (Ludwig). Bestimmung der Constante del

Pr.icession und der eigenen BewegungdesSonnen-
Bysteme.

Mem, de l'Acad. Imp. des Sci. de St.-P^ters-
bonrg, 1887, tome 35, No. 3, ::» pp.

ABERRATION AND HTUTAIION.

87. Airy (Sir Geo. Biddell). On a supposed alteration

in the amount of astronomical aberration of light,
produced by the passage of the lijjht through a
considerable thickness of refracting medium.

Proc. Roy. Soc, 1871, vol. 20, pp. 35-39; 1-73.
vol. 21, p. 121; Greenwich Asti lomical Ob-
servations, 1871, pp. cxii, cxix-exxxii (1 plate),
and 59-61.

88. Auwers (A). Ueber den Werth der Aberrations-

Constante nach den Beobachtungen von Moly-
neux.

Monatsberichte der Koniglieh Preussischeu Aka-
demie der Wissenschaften zu Berlin, 1869, pp.
611-646.

89. Ball (L. de). Bestimmung der Xutationsconstante.

Ast. Nach., 1885, Bd. 112, S. 33-42.

90. Brinkley (Rev. Dr. John). An account of observa-

tions made with the eight feet astronomical circle
at the observatory of Trinity College, Dublin,
since the begiuning of the year 1818, for investi-
gating the effects of parallax and aberration on
the places of certain fixed stars; also the compari-
son of these with former observations for deter-
mining the effects of lunar nutation.
Phil. Trans., 1821, pp. 327-360.

91. Busch (A. L.). Untersuchung der Bradley'achen in

Kew und Wansted angestellten Beobachtungen
zur Bestimmung der Aberration und Nutation.
Ast. Nach., 1836, Bd. 13, S. 329-338.

92. Downing (A. M. W. ). Discussion of the observa-

tions of ) Draconis, made with the Greenwich
rerlex zenith tube during the years 1857 to 1875,
inclusive.
Mouth. Not., 1882, vol. 42, pp. 325-344.

93. Gill (David). Notes on Nyreh'a determination of

the constant of aberration.

Note to accompany correspondence on the subject
of Nyr6u's determination of the constant of aber-
ration.

Mouth. Not., 18-4, vol. 44. pp. 275-277 and 2c!2.

94. Hall (Asaph). The constant of aberration.

The Astronomical Journal (Boston, Mass.), 1888,
vol. 8, pp. 1-5 and 9-13.

95. Henderson (Thomas). On the parallax of Siriua.

Mem. Roy. Ast. Soc. 1839, vol. 11. pp. 239-248.

96. Henderson (Thomas). The parallax of a Ceutanri,

deduced from Mr. Maelear's observations at the
Cape of Good Hope, in the years 1839 and 1840.
Mem. Rov. Ast. Soc. 1842, vol. 12, pp. 329-372.

152

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

97. Kiistner (F.) Neue Methode zur Bestimmung der

Aberrations-Constaute, nebst Untersuchuugeu
iiber die Yeriinderlichkeit der Polhohe.

Astronom. Beob. auf der Koniglichen Stern warte
zu Berlin, 1888, 59 pp. Reviewed by J. Bausch-
inger in V. J. Ast. Gesell., 1888. Bd. 23, pp. 251-
262.

98. Lindenau (Bemhard von). Versuch einer neuen

Bestimmung der Nutations- und Aberrations-
Constanten aus beobachteten Geraden-Aufsteiguu-
gen des Polaris.

Abbandlungen der Kdniglichen Akad. der Wis-
senschaften zu Berlin, 1841, math. Abhand., pp.
1-63.

99. Lindhagen (Daniel George). De numero constante

aberrationia et parallaxi annua stellae polaris ex
ascensionibus rectis bujus stellae, quae in specula
Pulcovensi annis 1842-1844 tubo culminatorio Er-
teliano observatae sunt deductia.

Mem. de l'Aead. Imp. des Sci. de St.-Pe'ters-
bonrg, Sci. Math, et Phys, 1853, t. 5, pp. 323-356.
99|. Lundahl (Gustavus). Specimen acadcmicum. De
nurneris untationis et aberrationia consiantibns,
atqne de parallaxi annua stellae Polaris, quales
dedncantur ex declinatiouibus stellae Polaris Dor-
pati annis 1832-1838 observatis. Quod Tenia
amplissiiiuD Facaltatia Philosophies ad Imperi-
alem Alexaudream in Fennia Universitatem P.
P. Gustavus Lundahl phil. doctor, reapondente
Johanne Henrico Ekliif, satacundensi. In audit.
philos. die xxvjj Aprilis MDCCCXLIL II. A.
M. S. Helsingforsiae.

100. Macleai (Sir Thomas). Determinations of the

parallax of a1 and afl Centauri, from observations
made at the Royal Observatory, Cape of Good
Hope, in the years l-<42-'43-'44, and 1848.
Mem. Roy. Ast. Soc, 1851, vol.20, pp. 70-98.

101. Maclear (Sir Thomas). Investigation of the paral-

lax of ft Centauri, from observations made at the
Royal Observatory, Cape of Good Hope.

Mem. Roy. Ast. Soc, 1852, vol. 21, pp. 141-152.

102. Main (Rev. Robert). On the values of the con-

stants of nutation aud aberration, and of the
parallax of y Draconis, as deduced from the obser-
vations made with the twenty-five-foot zenith
tube at the Royal Observatory, Greenwich.
Mem. Roy. Ast. Soc, 1855, vol. 24, pp. 147-187.

103. Main (Rev. Robert). On the value of the constant

of aberration as deduced from eight years' obser-
vations of y Draconis, from 1852 to 1859, inclusive,
made witli the reflex zenith tube at the Royal
Observatory, Greenwich.

Mem. Roy. Ast. Soc, 1859-'60, vol. 29, pp.
169-190.

103*. Maxwell ( Jas. Clerk). On a dynamical top, for
exhibiting the pbenomenaofthe motion of a system
of invariable form about a fixed point, with some
suggestions as to the earth's motion.

Trans. Roy. Soc. of Edinburgh, 1857, vol. 21,
pp. 559-570.

104. Nyren (Magnus). Bestimmung der Nutation der

Erdachse.

Mem. de l'Aead. Imp. des Sci. de St.-P6ters-
bourg, 1S72, tome 19. No. 2, 64 pp.

105. Nyr^n (Magnus). Die Polhohe von Pulkowa.

Mem. de l'Aead. Imp. des Sci. de St.-P6tera-
bourg, 1873, t. 19, No. 10, 40 pp.

106. Nyren (Magnus). L'aberratjon des e'toiles fixes.

Meni. de l'Aead. Imp. des Sci. de St.-P6tersbourg,
1883, t. 31, No. 9, 47 pp.

107. Nyren (Magnus). Remarqnes sur les "Notes on

Nyre'n's determination of the constant of aberra-
tion " de M. Gill.
Mouth. Not., 1884, vol. 44, pp. 277-282.

108. Nyren (Magnus). Zur Aberratiou des Fixsterne.

Melanges mathematiquea et astronomiqnes du
Bullelin de l'Acaddmie de St.-P6tersbour<;, 1888, t.
6, pp. (i")3-0G7; Bulletin Astronomique, 18.88, t. 5,
pp. 434-435.

109. Peters (C. A. F.). Numerus constans nntationisex

ascensionilms rectis stellae polaris in specula Dor-
pateusi annis 1822 ad 1838 observatis dednctns.
Adjecta est disquisitio theoreticade formula unta-
tionis.

Mem. de l'Aead. Imp. des Sci. de St.-P6ters-
bourg, Sci. Math, et Phys., 1-44, t. 3, pp. 125-227.
109^. Peters (C. A. F.). L'ber die Gesetz der Rotation
der Erde (dieae ale einem Korper mit 2 gleichen
Hauptaxen anaehend) urn cine dnrch den Schwer-
punct gehende Axe, die nieht mit der dritten
Hauptaxe znaammenfallt.
Ast. Nach., L844, lid. 22. s. 124-12^.

110. Peters (C. A. F.). Resultate ana Ueobachtungcn

des Polarsterns am Ertelechen Verticalkreiae der
I'ulkowaer Sternwarte.

Ast. Nach., L844, Bd. 22, S. 71-9G, 103-10*,
119-128. See aho Meni. de l'Aead. Imp. des Sei. de
St.-P6tersbonrg, Sci. Math, et Phys, 1853, t. 5, pp.
119-122.

111. Peters (C. A. F. ). Recherches sur la parallaxe des

gtoilea fixes.

M6m. de l'Aead. Imp. des Sci. de St.-P6ters-
bourg, Sci. Matb.et Phys.. 1853, tome ."., pp. 1-180

112. Richardson (W). Determination of the constant

of the aberratiou of light, from 4119 observations
made at the Royal Observatory, Greenwich,
during the years 1825, 1820, 1827. aud 1828, with
the two mural circles of Troughton aud Jones.
Mem. Roy. Ast. Soc, 1828, vol. 4, pp. 59-127.

113. Robinson (Rev. Dr. T. R.). On the constant of

lunar nutation, as given by the mural circle ob-
servations of Mr. Pond, from 1812 to 1835.
Mem. Roy. Ast. Soc, 1838, vol. 11, pp. 1-19.

114. Stone (B. J). A determination of the constant of

nutation from the observations in N. P. D. of Po
laris, Cephei 51, and S Urs.-e Minoris, made with
the transit circle of the Royal Observatory,
Greenwich, 1851-1865. Addendum. A determi-
nation of the constant of nutation from the
observations of Polaris in R. A. made at the
Royal Observatory, Greenwich, from 1851-1>ij7.
Mem. Roy. Ast. Soc, 1868, vol. 37, pp. 75-249.

115. Struve (F. G. W.). De numero constanti aberra

tionis et parallaxi annua fixarum ex observatio
nibus stellarum circumpolarium in ascensioue
oppositarum.

<>\ THE SOLAR PARALLAX AND ITS RELATED CONSTANTS

153

115. Struve (P. G. W.)— Continued.

F. G. W. Struve. Observationes astronomicas,
institutas iu specula Universitatis cacsareae Dor-
patensia, pablici juris faeit Senatus Universitatis-
Vol. 3, Observationes annorum 1820 et 1821, pp.
li-lxxxx. (Dorpati, 1822, - by 9f inches.)

116. Struve (F. G. W.). Sur les coustantes de l'aber-

ration et de la nutation.

Ast. Nach., 1841, Bd. 18. S. '269-294.

117. Struve (P. G. W. Rapport sur le m6moire de M.

C. A. F. Peters: Nnmerns constans nutationis ex
aseensionibusrectis stellae polaris in specula Dor-
patensi ab auuo 1822 ad 183b observatis dednctus.
Adjecta est disquisitio theoretica de formula
nutationis.

Acad, de St.-Petersbonrg. Bull. Sci., 1-4-'. t. 10,
cols. 145-160. Also Ast. Nach., 1843, Bd. 21, S.
81-94.
11-. Struve (F. G. W.). Snr le coefficient constant
dans l'aberration des etoiles fixes deduit des ob-
servations qui ont ete executees a l'observatoire
de Poulkova par l'iustrnuient des passages de
Repsold etabli dans le premier vertical.

Mem. de l'Acad. Imp. des Sci. de St.-Peters-
bonrg, Sci. Math, et Phys., 1844, t. 3. pp. 229-285.

119. Struve (Otto). Snr l'exactitude qui doit et re at-

tribute a la valeur du coefficient constant de
['aberration, deteranne'e a Poulkova.
Comptes Rendns, 1872, t. 75. pp. 795-798.

120. Villarceau (Yvon). Sur la constaute de l'aberra-

tion et la Vitesse de la luniiere. considerees dans
leurs rapports avec le monvement absolu de trans-
lation do systeme solaire.
Comptes Bendus, 1-72, t. 75, pp. 854-860.

121. Villarceau (Yvom. Recberches sur la the"orie de

['aberration, et considerations sur l'iurluence du
moavement absolu du systeme solaire, dans le
phenomene de l'aberration.
Comptes Rendna, 1-75, t. 81, pp. 165-171.

122. Villarceau Tvon). Theorie de l'aberration, dans

laqoelle U est teun eompte du monvement du
systeme solaire.

Connaissance des Temps. 1878. Additions, pp.
3-67.

ECLIPSES OF JVriTER S SATELLITES.

123. Bailly (J. S.). Meinoire sur les iuegalitesde la lumiere

des satellites de Jupiter, sur la mesure de leurs
diametres; et aur nn moyeu. aussi simple que
commode, de rendre les observations compara-
ble*, en reniediaut a la difference des vues et des
lunettes.

Hist, de l'Acad. Roy. des Sci. (Paris), 1771,
If 6m., pp. 5-0-6< 7.

124. Bailly (J. S. . A letter to the Eev. Nevil Maskelyne,

F. R. S., Astronomer Royal, from Mr. Bailly, of tbe
Royal Academy of Sciences at Paris: containing
a proposal of some new methods of improving the
theory of Jupiter's Satellites. Translated from
the French, with the original underneath.
Phil. Trans., 1773. pp. 1-5-216.

125. Cornu (A.). Snr la possibility d'accroitre dans nne

grande proportion la precision des observations
des eclipses dea satellites de Jupiter.
Comptes Rendus, 1-83. t. 96, pp. 1609-1615.

1-' Cornu (A.) et Obrecht (A). Eludes experimeo-
mentalee relatives a ['observation photo'ii6trique
des eVIip-.es des satellites de Jupiter.
Comptes Rendus, 1883, t. 96, pp. 1815-1821.

127. Delambre (J. B. J }. Tables eoliptiques des satel-

lites de Jupiter, d'apres la tbeorie de M. le Mar-
quis de La Place, el la totality <!• « observations
faitea depuis 1662 jusqu'a l'au 1-02: par M. Del-
ambre. * * Pans, 1-17. 4to, pp. lviij-f-135.

128. Downing (A. M. W). The eclipses of Jupiter's

satellites. [Being an account of Glasenapp's in-
vestigation of the eclipses of Jupiter's satellites,
and the light equation.]

The Observatory, (London), 1889, vol. 12, pp.
173-176 and 210-213.

129. Glasenapp (S.). Investigation of the eclipses of

Jupiter's satellites. A dissertation for tbe degi
of master of astronomy, by S. asenapp. B
Petersburg, 1S74. 8vo, pp. iv+158. (In the Rus-
sian language.)
129}. Obrecht (A.). Etude sur les eclipses dea satellite*
de Jupiter.

Annales de l'Observatoire de Paris, Me'moires, t.
18, pp. H. 1-H. 100. (Paris. 1-

TLTE VELOCITY OF LIGHT

130. Cornu (A.). Determination nouvelle de la vit

de la lumiere. ( 1 plate. )

Journal de l'Ecole Polytechniqne public* par le
Conseil dTnstruction de cot etablissement. 44^n«
cahier, tome 27. pp. 133-180. Paris, 1-74. 4to.

131. Cornu (A.). Memoire sur la determination de la

vitesse de la lumiere eutre l'Observ toire (de
Paris) et Montlhery. (7 plates). 315 pp.

Annales de l'Observatoire de Pari-, Me'moires,
tome 13, pp. A.1-A.315. Paris. 1-76. 4to.

132. Fizeau (H L.). Snr nne experience relative a la

vitesse de propagation de la lumiere.
Comptes Rendu-. 1-40. t. 29, pp. 90-92.

133. Foucault (Leon). Determination experiment ale

de la vitesse de la lumiere ; parallaxe du soleil.

Comptes Rendus, 1662, tome 55, pp. 501-503.
Also Recneil des Travaux Scientifiques de Leon
Foucault, pp. 216-218.
134 Foucault (Leon). Determination experimentael
de la vitesse de la lumiere; description des ap-
pareils.

Comptes Rendus, 1862. tome 55. pp. 792-796.
Also Recueil des Travanx Scientitiuues de L.
Foucault, pp. 219-226 and 546-54-. [2 plates and
many details not in Comptes Rendus. ]

135. Foucault (Leon ). Recueil des travaux scientifiques

de Leon Foucault ■ * * Public par Madame
Veuve Foucault sa mere. Mis en ordre par C.-M.
Gariel * * * Et precede d'uue notice snr les
oenvres de L. Foucault par J. Bertrand. Paris.
1878. 2 vols., 4to. T. 1, pp. xxviij-f-592: t. 2. pp.
18 and 19, plates.

136. Helmert (F. R.). Ueber eine Andeutnug constanter

Fehler in Cornu's neuester Bestimmung der Lieht-
geacbwindigkeit.

Ast. Nach., 1676, Bd. 87, S 123-126

I1

i54

< >N THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

137. Michelson (Albert A.). Experimental determina-

tion of the velocity of light. (12 figs.)

Proceed, of the Amer. Association for the Ad-
vancement of Science, 1679, pp. 124-160.

138. Michelson (Albert A.). Experimental determina-

tion of the velocity of light, made at the U. S.
Naval Academy, Annapolis. (June and July,
1879.) (15 figures.)

Astronomical Papers prepared for the use of the
Amer. Ephem. and Nautical Almauac, vol. 1, pp.
109-145. (Washington, 1862. 4to.)

139. Michelson (Albert A.). Supplementary measures

of the velocities of white and colored light in air,
water, and carbon disulphide * * *. (3 figs.)
Ast. Papers of Amer. Ephem., vol. 2, pp. 231-258.

140. Newcomb (Simon). Measures of the velocity of

light made under direction of the Secretary of
the Navy during the years 1880-'82. (7 plates.)
Ast. Papers of Amer. Eph., vol. 2, pp. 107-230.

141. Young (Jas.) anil Forbes (G.). Experimental

determination of the velocity of white and of col-
ored light.
Phil. Trans., 1882, pp. 231-289.

MASSES OF THE MOON AND PLANETS.

142. Airy (Sir Geo. B.). On the corrections in the ele-

ments of Delambre's solar tables required by the
observations made at the Royal Observatory,
Greenwich.
Phil. Trans., 1828, pp. 23-34.

143. Airy (Sir Geo. B.). Continuation of researches

into the value of Jupiter's mass.
Mem. Roy. Ast. Soc, 1838, vol. 10, pp. 43-47.

144. Asten (E. von). Resultate aus Otto von Struve's

Beobachtungen der Uranustrabanten.

Me"m. de PAcad. Imp. des Sci. de St.-P6ters-
bourg, 1872, t. 18, No. 5, 26 pp.

145. Asten (E. von). Untersuchungen fiber die Theorie

des Encke'schen Cometen. II. Resultate aus deu
Erscheinungen, 1819-1875.

Me"tn. de l'Acad. Imp. des Sciences de St.-Pe'ters-
bourg, 1876, tome 26, No. 2. 4to, 125 pp.

146. Backlund (O.). Comet Encke, 1865-1885.

Me"ui. de l'Acad. Imp. des Sciences de St.-P6ters-
bourg, 1886, tome 34, No. 8. 4to, 43 pp.

147. Bessel (Friedrich Wilhelm). Bestimmung der

Masse des Jupiter.

Astronomische Untersuchungen, Bd. 2, pp.
1-94. (Kiiuigsberg, 1842. 4to.)

148. Bessel (F. W.). [Mass of Saturn] in his paper

entitled " ZweiteFortsetzung der Untersuchungen
fiber die Bewegung der ifii^enischen Saturns-
Satelliten."
Ast. Nach., 1833, Bd. 11, S. 24.

149. Bond (G. P.). Beobachtungen auf der Sternwarte

in Cambridge (Nord-America) von dem Director
Herrn Professor Bond gemacht.
Ast. Nach., 1850, Bd. 31, S. 35-42.

150. Bouvard (A.). Tables astronomiques publiees par

le Bureau des Longitudes de France, contenant
les tables de Jupiter, de Saturne, et d'Uranus,
construites d'apres la theorie de la mecanique
celeste: par M. A. Bouvard * * Paris, 1821.

4to, pp. xxviij+110.

151. Burckhardt (J. C). Sur les masses des Plauetes.

Conn, des Terns, 1816. Additions, pp. 341-344.

152. Encke (J. F). Ueber die Bahn der Vesta.

Abhandlungen der Koniglichen Akademie der
Wissenschaften zu Berlin, 1826, Mathematische
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153. Encke ( J. F.). "Tafeldersuccessiven Anderungen

der Planeten-Massen" on p. 5 of his paper entitled
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Abhandlungen der Koniglichen \kademie der
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154. Encke (J. F.). [Masses of 7 principal planets.]

(A long list, with authorities.)

Ast. Nachr., 1812, Bd. 19, S. 187-190. (Not so
complete as Encke's table in the Abhandlungen
of the Berlin Akad., 1842, p. 5.)

155. Encke (J. F.). Sur la masse de Mercure.

Comptes Rendus, 1843, tome 16, p. 196; Ast.
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Berlin, Akad., 1842, Math. Abhandl., p. 56.
156 Encke (J. F.). Mass of Mercury ; in pp. 46 to 52 of
his paper, Uber den Cometen von Pons, 6te Ab-
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Abhand. der Koniglichen Akademie der Wissen-
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157. Haertl (E. De., otherwise E. Frhr. v. Haerdtl). Sur

l'orbite de la comete peYiodique de Winnecke et
sur une nouvelle determination de la masse de
Jupiter.

Comptes Rendus, 1888, tome 107, pp. 588-590;
Ast. Nachr., 1888, Bd. 120, S. 257-272.
157^ Haerdtl (Eduard Freiherrn von). Die Bahn des
periodiscben Kometen Winnecke in den Jahren
1858-1886, nebst eiuer neuen Bestimmung der
Jupitersmasse.

Denkschriften der Kaiserlichen Akademie der
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1889, Bd. 55, pp. 215-308.

158. Hall (Asaph). Observations and orbits of the

satellites of Mars. With data for epbemerides
in 1879. By Asaph Hall, Prof, of Math., U. S.
Navy. Rear-Admiral John Rodgers, U. S. Navy.
Sup't of the Naval Observatory. Washington :
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158£. Hall (Asaph). The orbit of Iapetus, the outer
satellite of Saturn.

Washington Observations, 1882, Appendix I.
4to, 82 pp.

159. Hall (Asaph). The orbits of Oberon and Titania,

the outer satellites of Uranus.

Washington Observations, 1881, Appendix I.
4to, 33 pp.

160. Hall (Asaph). Orbit of the satellite of Neptune.

Washington Observations, 1881, Appendix II.
4to, 27 pp.

161. Hall (Asaph, Jr.). Determination of the orbit of

Titan and the mass of Saturn.

Transactions of the Astronomical Observatory
of Yale University, vol. 1, part 2, pp. 107-148.
New Haven, 1889. 4to.

162. Hill (Geo. W.). [Mass of Venus] on pp. 2 and 36

of bis work entitled "Tables of Venus, prepared
for the use of the American Ephemeris and Nauti-
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the Navy. Washington, 1872. 4to. pp. 37+^1.

ON ["HE SOI \k PARALLAX AND ITS kl.l.A I ED < I >NSTAN fS

'55

103.

164.

104*

105.

166.

Ifi7.

168.

169.

170.

171.

172

173

174

17:

Hind (John Russell). Note upon the mass of the
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his satellite.

Month. Not , 1849, vol. 9, pp. 202-203.
Hind (J. R.). On the satellite of Neptune.

Month. Not.. 1654, vol. 15. pp. 46-48.
Kempf (P. ). Qntersnchungen iiher die Masse des
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Publicationcn des AstrophysikalischcnObserva-
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Krueger (A.). Untersuchung iiher die Balm des
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det Anziehung des Jupiter. Abdruck ans den
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La Grange (Joseph Louis). I'eber die Abnahnie
der Schiefe der Ecliptic.

Astronotnisches Jahrbuch, 1782, pp. 104-117.
(Berlin, 1779.)
La Grange (J. L.). [Masses of Mercury, Venus,
Earth, Mars, Jupiter, and Saturn] in his paper
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Nouveaux Mdmoires de I'Aeademie Roy ale des
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p. 190. Mso, pp. 181, 183, and 186.
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Mem. Roy. Ast. Soc, 1840, vol. 11, pp. 51-60.
Le Verrier (U. J.). Recherches sur l'orbite de
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de la masse de V6nus, et du diametre dn soleil.

Journal de Mathe'matiques Pureset Appliquees,
ou Recueil Mensuel de Memoires sur les Di verses
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Annales de l'Observatoire Imperial de Paris.
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Le Verrier (U. J.). Theorie et tables dn mouve-
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Annales de l'Observatoire Imperial de Paris.
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la paraUaxe dn Soleil.
Comptes Rend us, 1872, tome 75, pp. 165-172.
Lindenau (Bemhardo de). Investigate nova
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Lynn (W. T.). On the mass of Jupiter, as deduced
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Month. Not., 1866, vol. "27, pp. 1-5.
Moller (Axel). Bericht fiber die Resultate einer
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Vierteljahrsschrift der Astronomiscben GeseU-
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17)'). Newcomb (Simon). The (Iranian and Neptunian
Bystems, investigated with the 26-inch eqnatoi
of the United States Naval Obsen atory.

Washington Observations, 1-7:;, Appendix I.
71 pp.

177. Newcomb (Simon). Discussion and results ••!

observations on transits of Mercury, from it;77

to 1881.
Astronomical Papers prepared for the use of the

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pp.363-484. .Washington.!—,'. 4to.)
17-. Nicolai (F. B. G.). Die Elemente der Junobabn,

aut's neue berechnet, vom Hrn. Professor Nicolai

Direktor <ler Sternwarte in Mannheim unterm 11.

Sept. 1823 eingesandt.
Astronomisches Jahrbuch, 1826, pp. 224-228.

(Berlin. 1823.)

179. Peirce (Benj.). [On the mass of N /tune from

perturbations of Uranus.]

Ast. Nach., 1848, Bd. 27, S. 205.

180. Powalky (C). Bestimmung der Sonnenparallaxe

aus dem Verhaltniss der Erd- zur Sonnenmasse.
Ast. Nach., 1H72, Bd. 79, S. 25-28.

181. Powalky (C). Die Elemente der Sonnenbahn

und die Massen der Planeten Venus und Mars, aus
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Ast. Nach., 1876, Bd. 88, S. 257-276.

182. Earl of Rosse. Observations of the satellites of

the planet Uranus, made at Birr Castle, during
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Monti.. Not.. 1-75, vol.35, pp. 300-304.

183. Rothman (R. W.). Note ou the masses of Veuus

and Mercury.

Month. Not., 1842. vol. 5, pp. 130-133; Mem.
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1-4. Rothman (R. W.). Second note on the mass of
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Month. Not., 1-4-2. vol. 5, pp. 180-182; Mem.
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Month. Not., 1862, vol. 22, pp. 142-144.

186. Schur (W.). Bestimmung der Masse des Jupiter

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Vierteljahrsschrift der Astronomischen Gesell-
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187. Stone (E. J.). A determination of the moon's

mass.

Month. Not.. 1868, vol. 28, p. 42.
187$. Strnve (Hermann). Beobachtungen der Satan 8-
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Supplement I aux Observations de Poulkova.
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188. Struve (Otto). Determination de l'orbite dn satel-

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189. Tisserand (F.). Sur la determination des mas

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Comptes Rend us. 1881, t 92, pp. 653-660.

i56

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

190. Wurm ^ J. F.). Ueber die Masse des Uranus, nebst

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Astronomisches Jahrbuch, 1792, pp. 210-214.
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191. Wurm (J. F.). Versuch, die Masse der Venus aus

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Astronomisches Jahrbucb, 1806, pp. 149-l.r>3.
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THE .SOLAR PARALLAX FROM TRIOOXOHETRWAL
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L92. Airy (Sir Geo. B.). Report on the telescopic ob-
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A Parliamentary Return, ordered by the House
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193. Airy (Sir Geo. B.). On the inferences for the value

of mean solar parallax and other elements deduci-
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Month. Not., 1877, vol. 38, pp. 11-16.

194. Airy (Sir Geo. B.). Account of observations of the

transit of Venus, 1874, December 8, made under
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195. Bond (W. C). Solar parallax deduced from

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The Astronomical Journal, edited by Benj. Ap-
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196. Cassini (J. D.). Les elevens de l'astrouomie veri-

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197. Cruls (L.). Sur la valeur de la parallaxe du soleil,

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Comptes Rendus, 1887, t. 105, pp. 1235-1237.
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Ast. Nach., 1879, Bd. 96, S. 119-128.
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200. Eastman (J. R. ). The solar parallax from meridian

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Washington Observations, 1877, Appendix III,
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201. Encke (J. F.). Die Entfernung der Sonne von der

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202. Encke (J. F.). Der Vcnnsdnrchgang von 1769, als

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203. Encke (J. F. ). tiber den Vennsdurchgang von 1769.

Abhandlungeu der Konigliebeu Akademie der
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203$. Euler (L.). Expositio methodornm cum pro deter-
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Novi Commentarii Academiae Scientiarvm Im-
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204. Faye (H. A. E. A). Sur les passages de V<5nus et

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205. Faye (H. A. E. A.). Examen critique des id^es et

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206. Faye (H. A. E. A.). Sur la parallax dn soleil.

Comptes Rendus, 1881, t. 92, pp. 375-37-.

207. Faye (H. A. E. A.). Rdponse k qnelques critiques

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209. Flamstead (John). [The solar parallax from ob-

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210. Galle (J. G.). Ueber die An wendung von Beobach-

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211. Galle (J. G.). Ueber den aus den Flora-Beobacb-

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Ast. Nach., 1875, Bd. 85, S. 257-270: and 1875,
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212. Galle (J. G). Lettre de M. Galle, directeurde l'Ob-

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ON THE SOLAR PARALLAX AND ITS RELATED CONSTAT

'57

613.

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815.

221.

217.

218.

219.

220.

221.

222.

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224.

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Gill (David). Account of a determination of the
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Mem. Roy. Ast. Soc, 1880-81, vol. 4(5, pp. 1-172;
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Gill (David). Note on some criticisms made by Mr.
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Month. Not., 1883, vol. 43, pp. 307-315.

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The IT. S. Naval Astronomical Expedition to the
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Hall (Asaph). Solar parallax, deduced from ob-
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Washington Observations, 1863, Appendix A, pp.
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Hall (Maxwell). Determination of the solar par-
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Mem. Roy. Ast. Soc, 1877-79, vol. 44. pp. 51-121.

Halley (Edmund). Some remarks upon the
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Phil. Trans., 1720, pp. 113-116.

Hansen (P. A.). [Solar parallax derived from the
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Abhand. der mathem.-phys. Classe der Konig.
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Hansen (P. A.). On the value of the solar paral-
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Month. Not.. 1863, vol. 23, p. 242.

Hansen (P. A.). Calculation of the Sun's parallax
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Month. Not., 1863, vol. 24, pp. 8-12.

Harkness (Wm.). On the relative accuracy of
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Aiuer. Jour, of Science, 1881, vol. 22, pp. 375-394-

Heuderson (T.). Letter from Mr. Henderson to
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Hornsby (Thomas). A discourse on the parallax
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Phil. Trans., 1763, pp. 467-495.
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Phil. Trans., 1771, pp. 574-579.

228.

229.

227. La Lande (J. J. le Francois de). M6tnoire mi Id
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ft a la Chii

llistoire de I'Aoademie Boj ale i!« Sciences, 1771.
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Annalea de I'Observatoire Imperial de Paris,
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Le Verrier (U. J.). On the masses of the planets
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Mouth. Not., 1872, vol. 32, pp. 322-328. (This
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229$. Lexell (And. Joh.). Disqnisitio de investiganda
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NoviCommentariiAcadeniiaeScientiarum [mpe-
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230. Lord Lindsay«»rf Dr. David Gill. Determination

of the solar parallax by observations of the minor
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Dun Edit Observatory Publications, 1877, vol.
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231. Littrow (Carl Ludwig). P. Hell's Reise nach

VVardoe bei Lappland and seine Beobachtung des
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lauterungen begleitet von Carl Ludwig Littrow
* # * Wien, 1835. 12mo, pp. xvj+166.

232. Newcomb (Simon). Investigation of the dis-

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Washington Observations, 1865, Appendix II.
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233. Newcomb (Simon). Remarks on Mr. Stone's re-

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Month. Not., 1868, vol. 29, pp. 6-7.

234. Newcomb (Simon). On Hell's alleged falsilica-

tiou of his observations of the transit of Venus
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Month. Not., 1883, vol. 43, pp. 371-381.
234$. Newcomb (Simon). Discussion of observations
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Astronomical papers prepared for the use of the
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Obrecht (A.). Discussion des resultats obteuns
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Comptes Rendus, 1885, t. 100, pp. 227-230.

Obrecht (A.). Sur la parallaxe sol aire de'duitc des
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Comptes Rendus, 1885, t. 100, pp. 341-343.

237. Obrecht (A). [Definitive value of solar parallax
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Comptes Reudus, 1885, t. 100, p. 1121.

235.

236.

158

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

238. Pingre (Alexandre Guy). Observations astrono-

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Histoiredel'AcademieRoyale des Sciences, 1761.
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239. Pingre (A. G.). Nouvelle recherche sur la ddtermi-

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Histoiredel'AcademieRoyale des Sciences, 1765.
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240. Pingre (A.G.). Mcmoire sur la parallaxe du soleil,

d6duite des rneilleurs observations de la dur<5e du
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Ilisioiredel'Acade'inieRoyaledes Sciences, 1772.
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24 1 . Plana (Jean). Surla parallaxe du soleil et la masse

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Theorie du mouveruent de la lune, par Jean
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Phil. Trans., 1768, pp. 109-127.

243. Pound (Rev. James). Nupera observationes astro-

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Phil. Trans., 171. i, pp. 1109-1114.

244. Powalky (C). Discussion nouvelle du passage

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Conuaissance des Temps, 1867. Additions, pp.
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245. Powalky (C). Beitriige zu einer vollstiindigeren

Beurtheilung der Veunsdurchgjinge und Ermit-
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Ast. Naeh., 1870, Bd. 76. 8. 101-192; and 1-71.
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24(5. Powalky (C). Ueber die verschiedeneu Metho-
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247. Puiseux (V. A.). Snr les observations de contact
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Comptes Rendu*. 1881, t. 92. pp. 4^1-488.

24-. Puiseux (V. A.). Sur les mesores ndcrom^triques
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Comptes ^eudus, 1381, t. 92, pp. 808-813.

249. Short (James). The observations of the internal

contact of Venus with tbe Sun's limb, in the late
transit, made in di tic nut places in Europe, com-
pared with the time of the •.aim- contact observed
at the Cape of Good Hope, and the parallax of
the Sun from thence determined.
Phil. Trans., 1702, pp. 011-02-.

250. Short (James). Second paper concerning the par-

allax of the Sun determined from the observations
of the transit of Venus, in which this subject is
treated of more at length, and the quantity of the
parallax more fully ascertained.
Phil. Trans., 1763, pp. 300-345.

251. Stokes, Adams, Christie, Hind, Richards, Stone,

Dove. Report of the committee appointed by
the British Government to superintend the ar-
rangements to be made for the sending of expedi-
tions at the Government expeuse, and securing
cooperation with the Goverumeut expeditions for
the observation of the transit of Veuus, 1882, De-
cember 6.

A Parliamentary Return, June, 1887. 16 by 8i
inches. 88 pp.

252. Stone (E. J.). A determination of the Sun's mean

equatorial horizontal parallax, from declination ob-
servations of Mars and stars, made during the op-
position of 1^62, at the Royal Observatory, Green-
wich ; the Royal Observatory, Cape of Good Hope ;
and the Government Observatory, Williamstown,
Victoria.

Mem. Roy. Ast. Soc, 1863-64, vol. 33, pp. 77-102.

253. Stone (E. J.). Note on the calculation of the Suu's

parallax from the lunar theory by P. A. Hansen.
Month Not., 1867, vol. 27, pp. 239-240.

254. Stone (E. J.) Determination of a slightly corrected

value of the solar parallax from the data of Le
Verrier's solar rabies.
Month. Not., 1867, vol. 27, pp. 241-242.

255. Stone (E. J.). Some remarks on the value of the

solar parallax, as deduced from the parallactic in-
equality in the earth's motion.

Month. Not., 1807, vol.28, pp. 21-25.

256. Stone, (E. J.). A rediscussion of the observations

of the transit of Venus, 1769.

Month. Not., 1868, vol. 28, pp. 255-266.

257. Stone (E. J.). A reply to Mr. Newcomb's remarks

on "Mr. Stone's rediscussion of the transit of
Venus, 1769."
Mouth. Not., 1868, vol. 29, pp. 8-11.

258. Stone (E. J.). On some points connected with the

rediscussion of the observations of the transit of
Venus, 1769.
Month. Not.. 1809, vol. 29, pp. 236-249.

259. Stone (E. J.). On some effects of the comparative

clinging of the limb of Venus to that of the Sun

iu the transit of 1874, as compared with that of

1-82.
Month. Not., 1869, vol. 29, pp. 250-252.

260. Stone (E. J.). On some phenomena of the internal

contacts common to the transits of Venus, observed
iu 1709 and 1874, and some remarks thereon.
Month. Not., 1870, vol. 37, pp. 45-55.

261. Stone (E. J). On the telescopic observations of the

transit of Venus 1^74, and on the conclusions to
be deduced from these observations.
Month. Not., 1878, vol. 38, pp. 279-295.

202. Stone (E. J.). A comparison of the observations of
contact of Venus with the Sun's limbiu the transit
of 1874, December 8, made at the Royal Observa-
tory, Cape of Good Hope, with the corresponding
observations made at the stations in the Parlia-
mentary Report, and a dismission of the results.
Month. Not., 1878, vol. 38, pp. 341-347.

263. Stone (E. J.). Note on the durations observed by
the French observers at the transit of Venus, 1874.
Month. Not., 1881, vol. 41, pp. 325-329.

ON I III SO] \k PARALLAX AND I Is RELA fED CONSTANTS

'59

264. Stone (E. J.). On some results obtained from tho

meridian observations of Mars at the opposition
of 1877.

Month. Not., 1882, vol. 42, pp. 290-301.

265. Taylor (Thomas Glanville). [Solar parallax from

declinations of Mars observed at Madras and the
Cape of Good Hope during the years 1832 and 1834.]
Result of Astronomical Observations made at
the Hon. E. I. Company's Observatory, Madras.
Vol. 3. For the years 1834 aud 1835, p. 71. (Ma-
dras, 1836.)
Jiiii. Todd (David P.). The solar parallax as derived
from tin' American photographs of the transit of
Venn-. 1-7-1. December 8-9.

American Journal of Science, 1881, vol. 21, pp.
491-493.

267. Tnpman (Capt. G. L.). On the mean solar parallax

as derived from the observations of the transit of
Venus. 1874.

Month. Not., 187H, vol. 38, pp. 429-457.

268. Tnpman (Capt. G. L.). On the photographs of the

transit of Venus.

Month. Not., 1878, vol. 38, pp. 503-513.

269. Winnecke(A.). [The solar parallax from meridian

observations of Mars duriug the opposition of
1862. ]
Ast. Nach., 1863, Bd. 59, S. 261-264.

DE.\SITT OF THE EARTH.

270. Airy (Sir Geo. B.). Account of pendulum expe-

riments undertaken in the Harton Colliery, for the
purpose of determining the mean density of the
earth.

Phil. Traus., 1856, pp. 297-355.

271. Baily (Francis). An account of some experiments

with the torsion-rod, for determining the mean
density of the earth.

Mem. Roy. Ast. Soc, 1842, vol. 14. 4to, pp.
120+uc.\lvii.j.

272. Bianco ' Ottavio Zanotti). Ilproblema meccauico

della tigura della terra, esposto secondo i migliori
antori, da Ottavio Zanotti-Bianco * * * Parte
primo. Firenze-Torino-Roma : 1880. pp. xiv+302.
Parte sccouda, libro primo, 1885, pp. viij-f-188.
Bvo.

273. Callandreau (O.). Snr la theorie de la figure des

planetes.
Comptes Rendus, 1884, t. 99, pp. 1060-1061.

274. Callandreau (O.). Sur la constitution interieure

de la terre.

Compter Reudns, 1885, t. 100, pp. 37-40.
27.">. Callandreau (O.). Addition a deux notes pr6c6-
deutes, concernant la theorie de la figure des
planetes el de la terre.

Comptes Rendus, 1885, t. 100, pp. 163-161.

276. Callandreau (O.). Sur la th<5orie de la figure de la

terre.
Comptes Reudns, 1885, t. 100, pp. 1204-1206.

277. Callandreau (O.). Remarquea sur la theorie de la

figure ile la terre.

Bulletin Astrouomique (Paris), 1888, t. 5, pp.
473-480
27s. Callandreau (O.). Remarques sur la th6orie de la
figure de la terre.

Bulletin Astronotuique, 1389, t. 6, pp. 185-192.

279. Carlini (Francesco). Osservazioni della lunghezza

del pendolo semplioe fatte all'altezzadi null

sill livello del mare.

Efiemeridi Astronomiche di Milano. 1-21. Ap-
peudice, pp. 2-- 10.

280. Cavendish (Henry). Experiments to determine

the density of the earth.

Phil. Trans., 1798, pp. 469-626.

281. Clairaut (A. C). Theorie de la figure de la terre,

tiive des principes de I'hydrostatiqne, Par M.
Clairaut. ' " * Paris, 1743. l2mo, pp. xl-f 310.

282. Clarke (Capt. A. R.). On the deflection of the

plumb-line at Arthur's Seat, aud the mean speci-
fic gravity of the earth.

Phil. Trans., 1856, pp. 591-606.

28:?. Cornu(A.)<< Bailie (J. B.). Determination non-
velle do la constants de l'attraction ct de la den-
sity moyenne de la terre.
Comptes Rendns, 1873, t. 76, pp >4-958.

284. Cornu (A.) et Bailie (J. B.). Sur la niesare do la

densiti: moyenne de la terre.

Comptes Rendus, 1878, t. 86, pp. 699-702.

285. Darwin (Geo. H. ). On an oversight in the Mecaui-

qne Celeste, and on the internal densities of the
planets.

Month. Not., 1876, vol. 37, pp. 77--:'.

286. Darwin (G. H). On the figure of equilibrium of a

planet of heterogeneous density.

Proc. Roy. Soc, 1883, vol. 36, pp. 158-166.

286k. Fisher (Rev. Osmond). Physics of the earth's
crust. By the Rev. Osmond Fisher. * " * Sec-
ond edition, altered and enlarged. London, 1889.
8vo, pp. xvj+391.

287. Giulio (Charles Ignace). [Note] sur la determi-

nation de la densite moyenne de la terre deduite
de l'observatiou do peudule faite a l'Hospice du
Mont Cenis par Mr. Carlini en septembre 1821.

Memorie. della Reale Aecaderuia delle Scienze di
Torino, 1840, serie 2, tomo 2. Scienze hsiche e
inatheniatiche, pp. 379-384.
2874. Hansen (P. A.). [Die nurnerische Bestiromung
der Triigbeitsmomeute der Erde. ]

Ast. Nach., 1842, Bd. 19, S. 197-198.

288. Haughton (Rev. Samuel). On the density of the

earth, deduced from the experiments of the As-
tronomer Royal, in Harton coal-pit.

L., E. aud D. Phil. Mag., 1856. vol. 12, pp. 50-51.

289. Hill (G. W . i. On the interior constitution of the

earth as respects density.

Annals of Mathematics, edited by Prof. Oruiond
Stone, University of Virgiuia, 1888, vol. 4, pp.
19-29.

290. Hutton (Chas.). An account of the calculations

made from the survey and measures taken at
Schehallien, iu order to ascertain the mean density
of the earth.

Phil. Trans., 1778, pp. 689-788

291. Hutton (Chas.). The mean density of the earth.

Being an accouut of the calculations made from
the survey aud measures taken at Mouut Shichal-
lin, in order to ascertain the mean density of the
earth. Improved from the Philosophical Trans-
actions, vol. 68, for the year 1778.

i6o

ON IIII'. soi.AR PARALLAX AND ITS RELATED ((INSTANTS.

291. Hutton (Chas.) — Continued.

Tracts on mathematical and philosophical sub-
jects. » * - By Chas. Hutton, ix. i>. and F. R.
s., etc. Vol. 2, tract 26, pp. 1-63. London, 1812,
12nio. 384 pp.

292. Hutton (Charles). On the mean density of the

earth.

Phil. Trans., 1821, pp. 276-292.
29:5. Jolly (Ph. von). Die Auwendnng der Waage auf
Probleme der Gravitation. Zweite Abhandlang.

Abhandlangen der Mathematisch - l'hvsikali-
scbeu Classe der Kouiglich Bayerisehen Akadeuiie
der Wissenschaften, 1883, Bil. 14, 2te Abtheilun^.
pp. 1-26.

294. Le Gendre (A. M.). Suite des recherches sur la

figure des planetes.

Histoire de l'Acadeiuie des Sciences (Paris),
1789. pp. 372-454.

295. Levy (Maurice). Sur la th6orie de la figure de la

terra.

Comptes Rendus, 1888, t. 106, pp. 1270-1276,
1314-1319, and 1375-1381.

296. Lipschitz (R. }. Versuch zur Herleitung eiues

Gesetzes, das die Dichtigkeit fiir die Schichten ini
Inueru der Erde anuiihernd darstellt, ausden gege-
benen Beobachtuugeu.

Journal fiir die reiue und angewaudte Mathe-
matik (Borchardt. Berlin), 1863, Bd. 62, pp. 1-35.

297. Lipschitz (R.). Beitrag zur Theorie des Gleich-

gewichts eiues nicht honiogenen lltlssigen rotireu-
den Spharoids.

Journal fiir die reiue und angewaudte Mathe-
inatik (Borchardt, Berlin), 1863, Bd. 63, pp.
2>9-295.

298. Maskelyne (Rev. Nevil) An account of observa-

tions made on the mountain Schehallien for find-
ing its attraction.
Phil. Trans., 1775, pp. 500-542.

299. Mendenhall (T. C). Determination of the ac-

celeration due to the force of gravity, at Tokio,
Japan.
Amer. Jour, of Science, 1880, vol. 20, pp. 124-132.

300. Mendenhall (T. C). On .t determination of the

force of gravity at the summit of Fujiyama,
Japan.

Amer. Jour, of Sience, 1^81, vol. 21, pp. 99-103.

301. Plana (Jean). Note sur la deusite" moyenne de

l'ecorce superticielle de l-i terre.

Ast. Nach., 1852, Bd. 35, S. 177-191.

302. Playfair (John). Account of a lithological survey

of Schehallien, made in order to determine the
specific gravity of the rocks which compose that
mountain.
Phil. Trans., 1811, pp. 347-377.

303. Poincare (H.). Sur la figure de la terre. (On the

relation between interior density, flattening, Clai-
raut's equation, and precession.)
Comptes Rendus, 1888, t. 107, pp. 67-71.

304. Poincare (H). Sur la figure de la terre.

Bulletiu Astrouomique, (Paris) 1889, t. 6, pp.
5-11, and 49-60.

305. Poynting (J. H.). On a method of using the bal-

ance with great delicacy, and on its employment
to determine the mean density of the earth. (1
plate.)

Proc. Roy. Soc, 1878-9, vol. 28, pp. 2-35.

306. Radau(R.). Sur la loi des densit6s a l'iut<5rieur

de la terre.

Comptes Rendus, 1885, t. 100, pp. 972-974.

307. Radau (R.). Remarques sur la th6orie de la figure

de la terre.

Bulletiu Astronomique, 1835, t. 2, pp. 157-161.

308. R[adau] (R.). Revue des— Sterneck (R.)— Unter-

snchungen iiber die Schwere im Innern der Erde.
Bulletiu Astronomique (Paris), 1887, t. 4, pp.
234-237.

309. Reich (P.). Versuche iiber die mittlero Dichtig-

keit der Erde. inittclst der Drehwa^c von F.Reich,
Prof, der Physik an der K. S. Bergakademie. Mit
2 lithographirten Tafeln. Freiberjr, 1838. 12mo,
66 pp. [I have never seen this work, but on ac-
count of its historical importance I quote the title
from 271, p. 9.— W. H.]

310. Reich (P.). Neue Versuche mit der Drehwage zur

Bestimmung der mittleren Dichtigkeit der Erde.

Abbaudluugen der matbematisch-physischeu
Classe der Koniglich Sachsischen Gesellschaft der
Wissenschaften zu Leipzig, 1853, Bd. l,pp. 383-430.

311. Roche (Edouard). Mdmoire sur l'e'tat interieur du

globe terrestre. Par M. fidouard Roche * * *
Paris, 1881. 4to, 43 pp. (Extrait des Mdmoires
de l'Acad6mie des Sciences et Lettres de Montpel-
lier. Section des Sciences, tome 10.)

312. Roche (Edouard) Note sur la loi de ladensit6 a

l'iuterieur de la tern-.

Comptes Rendus. 1854, t. 39, pp. 1215-1217.

3124. Schell (Anton). Leber die Bestimmung der mitt-
leren Dichtigkeit der Erde, vou Anton Schell,
Professor am Baltischen Polytechuicum zu Riga.
Mit drei lithographirten Tafeln. Goettingen, 1809.
4to, 39 pp.

313. Sterneck (Robert von). Uutersuchungen fiber

die Schwere ini Innern der Erde.

Mittheilungen des kaiserl. konigl. Militiir-Geo-
graphischen Institutes (Wien), 1882, Bd. 2, pp.
77-120.

314. Sterneck (Robert von). Wiederholung der Un-

tersuchungen fiber die Schwere im Innern der
Erde.

Mittheilungen des kaiserl. konigl. Militar-Geo-
graphischen Institutes (Wien), 1883, Bd. 3, pp.
59-94.

315. Sterneck (Robert von). Untersuckungen fiber

die Schwere auf der Erde.

Mittheilungen des kaiserl. konigl. Militiir-Geo-
graphischeu Institutes (Wien), 1884, Bd. 4, pp. 89-
155.

316. Stieltjes. Note sur la density de la terre.

Bulletin Astronomique (Paris), 1884, t. 1, pp.
465-467.

317. Tisserand (P.). Quelques remarques au sujet de la

theorie de la figure des plauetes.

Comptes Reudus, 1884, t. 99, pp. 399-403 and 518.
318- Tisserand (P.). Sur la theorie de la figure de la
terre.

Comptes Rendus, 1884, t. 99, pp. 577-583.

319. Tisserand (P ). Quelques remarques au sujet de la
theorie de la figure des planetes.

Bulletin Astronomique (Paris), lb84, t. 1, pp.
417-420.

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

[6l

320. Tisserand (F.). Sur la constitution interioure do

la terre.
Bulletin Astronomiquo (Paris), 1884, t. 1, pp.
521-527.

321. Todhunter (I.). A history of the mathematical

theories of attraction, and the figure of the earth,
from the time of Newton to that of Laplace. By
I. Todhunter ■ * * London, 1873. 2vols.,8vo.
Vol. 1, pp. xxxvj+476 ; vol. 2, pp. 508.

322. Wilsing (J.). Bestimmung der mittleren Dichtig-

keit der Erde mit Hiilfe cines Pendelapparates.

Publicationen des Astrophysikalischcn Obser-
vatoriunis zu Potsdam, Bd. 6, S. 31-128 (1887, Nr.
22); 2te Abhaudlung, Bd. 6, S. 129-192 (1889,
Nr. 23).

THE TIDES.

323. Airy (Sir Geo. B.). Tides and waves.

Encyclopaedia Metropolitana. 2nd Division,
Mixed Sciences, vol. 3 pp. 241*-396*. (London,
1842. 4to.)

324. Airy (Sir Geo. B.). On the laws of the tides on

the coasts of Ireland, as iuferred from an exten-
sive series of observations made in connection
with the Ordnance Survey of Ireland.
Phil. Trans., 1845, pp. 1-124.

325. Airy (Sir Geo. B.). On a controverted point in

Laplace's theory of the tides.
L., E. & D. Phil. Mag., 1875, vol. 50, pp. 277-279.
32G. Airy (Sir Geo. B.). On the tides at Malta.
Phil. Trans., 1878, pp. 123-138.

327. Band (Maj. A. W.) and Darwin (G. H.). Results

of the harmonic analysis of tidal observations.
Proc. Roy. Soc, 1885, vol. 39, pp. 135-207.

328. Baird (Maj. A. W.). A manual for tidal observa-

tions, and their reduction by the method of har-
monic analysis; with an appendix containing
auxiliary tables to facilitate the computations.
By Major A. W. Baird, R. E., F. R. S., * * *
London, 1886. 8vo, pp. vi+54-f xl.

329. Challis (Jas.). On the mathematical principles of

Laplace's theory of the tides.

L. , E. & D. Phil. Mag., 1875, vol. 50, pp. 544-548.

330. Darwin (G. H.). Tides.

Encyclopaedia Britaunica (ninth edition, Edin-
burgh, 1888), vol. 23, pp. 353-381.
330J. Darwin (G. H.). Second series of results of the
harmonic analysis of tidal observations.
Proc. Roy. Soc, 1889, vol.45, pp. 556-611.
330J. Darwin (G. H). On the harmonic analysis of
tidal observations of high and low water.
Proc. Roy. Soc, 1890, vol. 48, pp. 278-340.

331. Darwin and Turner. On the correction to the

equilibrium theory of tides. I. By Prof. G. H.
Darwin. II. By Mr. H. H. Turner.

Proc. Roy. Soc, 1886, vol. 40, pp. 303-315.

332. Evans (Captain F. J. O.) and Thomson (SirWm.).

On the tides of tbe southern hemisphere and of
the Mediterranean.

Report of the 48th meeting of tho British Asso-
ciation for the Advancement of Science, 1878, pp.
477-481.
33:?. Ferrel (Win.). The problem of the tides, with re-
gard to oscillations of the second kind.

The Astronomical Journal (Cambridge, Mass.),
1856, vol. 4, pp. 173-176.

G987 11

340.

341.

334. Ferrel (Wm.). Discussion of tides iii Boston

harbor.

Report of the Superintendent of the D. 8. Coast
Survey, 186ft Appendix 5, pp. 51 L02.

335. Ferrel (Wm.). On the moon's mass as deduced

from ;i discussion of the I ides of lioston harbor.

Report of the Superintendent of the r. s. Coast
Survey, 1*70. Appendix 20, pp. 190 199.

336. Ferrel (Wm.). Tidal researches. An appendix to

the II. S. Coast Survey Report for 1-7 1, but printed

separately. 4io, pp. xiii-j 268.
:*37. Ferrel (Wm.). On a controverted point in La
place's theory of the tides.

L., E. &, D. Phil. Mag., 1876, vol. 1, pp. 182
187.

338. Ferrel (Wm.). Discussion of tides in Penobscot

Bay, Maine.

Report of the Superintendent of the U. S. Coast
and Geodetic Survey, 1878. Appendix 11, pp.
268-304.

339. Ferrel (Wm.). Discussion of the tides of the Pa-

cific coast of the United States.

Report of the Superintendent of the U. S. Coast
and Geodetic Survey, 1882. Appendix 17, pp.
437-450.
Ferrel ( Wm.). Report on the harmonic analysis of
the tides at Sandy Hook.

Report of the Superintendent of the U. S. Coast
and Geodetic Survey, 1883. Appendix 9, pp. 247-
251.
Ferrel (Wm.). Description of a maxima and mini-
ma tide-predicting machine. (5 plates.)

Report of the Superintendent of the U. S. Coast
and Geodetic Survey, 1883. Appendix No. 10, pp.
253-27 2.
Ferrel (Wm.). Laplace's solution of the tidal
equations.

The Astronomical Journal ( Boston, Mass.), 1880,
vol. 9, pp. 41-44 ; and 1890, vol. 10, pp. 121-123.
Finlayson (H. P.). On the mass of the moon as
deduced from tho mean range of spring and neap
tides at Dover during the years 1861, 1864, lsu5,
and 1866.

Month. Not., 1867, vol. 27, p. 271.
344. Haughton (Rev. Samuel). Discussion of tidal
observations made by direction of the Royal Irish
Academy in 1850-51.

Transactions of the Royal Irish Academy, 1854,
vol. 23, Part II, pp. 35-140.
Haughton (Rev. Dr. Samuel). On the tides of
the Arctic seas. Parts 1 to vn.
Phil. Trans., 1863, pp. 243-272; 1866, pp. 639-
" 655; 1875, pp. 317-360; and 1878, pp. 1-16.
Haughton (Rev. Dr. Samuel). On the semi-
diurnal tides of tho coasts of Ireland, deduced
from tho Academy observations. Parts 1 to xi.

Transactions of the Royal Irish Academy (Dub-
lin), 1871, vol. 24, Science, pp. 195-211, and 253-
350.
347. Houzeau (J. C.) and Lancaster (A.). [Bibliog-
raphy of tides.] (Thcone des marees. La uiarfo
et la rotation du globe.)

Bibliographic g<m6rale de l'astronomie, t. 2,
cols. 626-636 and 1626. (Bruxelles, 1*82, 4to.)

342.

343.

345.

346.

,

l62

ON THE SOLAR PARALLAX AND ITS RELATED CONSTANTS.

348. La Place (P. S. de). Memoire sur le flux et le re-
' flux de la mer.

Meinoires de l'Acaddmie Royale des Sciences de
l'Institut de France, 1818, t. 3, pp. 1-90.

349. La Place (P. S. de). [Mass of the moon from ob-

servations of the tides at Brest.]

Trait6 de Mecanique Celeste, t. 5, livre 13,
chart. 3, sec. 10.

350. Lubbock (Sir John Wm.). On the tides in the

port of Loudon.
Phil. Trans., 1831, pp. 370-415; 1832, pp. 595-599.

351. Lubbock (Sir John Wm.). On the tides [at Brest

and London"].
Phil. Trans., 1832, pp. 51-55.

352. Lubbock (Sir John Wm.). Note on the tides [at

Brest, Plymouth, Portsmouth, Sbeerness, London
Docks, and St. Helena].

Phil. Trans., 1833, pp. 19-22.

353. Thomson (Sir Wm.), Roberts (Edward) and

Darwin (G. H.). Reports of the committees
appointed (by the British Association) forthc pur-
pose of promoting the extension, improvement,
and harmonic analysis of tidal observations.

Reports of the meetings of the British Associa-
tion for the Advancement of Science, 1868, pp.
489-510; 1870, pp. 120-151; 1*71, pp. 201-207;
1872, pp. 355-395; 1876, pp. 275-307; 1883, pp. 49-
117; 1884, pp. 33-35; 1885, pp. 35-60; 1886, pp.
40-58.

354. Thomson (Sir Wm.). Ou an alleged error in La-

place's theory of the tides.

L., E. & D. Phil. Mag., 1875, vol. 50, pp. 227-242.

355. Thomson (Sir Wm.). Note on the "oscillations of

the first species" in Laplace's theory of the tides.
L., E. & D. Phil. Mag., 1S75, vol.50, pp. 279-284.

356. Thomson (Sir Wm.). General integration of La-

place's differential equation of the tides.
L., E. & D. Phil. Mag., 1875, vol. 50, pp. 388-402.

357. Thomson (Sir Wm.). The tide gauge, tidal har-

monic analyser, and tide predictor.

Minutes of Proceedings of the Institution of Civil
Engineers (London), 1881, vol. 65, pp. 2-25. Dis-
cussion, pp. 26-72. (16 figures.)

358. Whewell (Rev. Wm.). Researches on the tides.

Series 1 to 14.

Phil. Trans., 1833, pp. 147-236; 1834, pp. 15-45 ;
1835, pp. 83-90; 1836, pp. 1-15, 131-147, :<89-:i41 ;
1837, pp. 75-85, 227-244; 1838, pp. 231-247; 1839,
pp. 151-161, 163-166; 1840, pp. 161-174, 255-272;
1848, pp. 1-29; 1850, pp. 227-233.

THE ELECTRIC CONSTANT, v.

359. Kohlrausch (R.) and Weber (W.). Elektrodyna-

inische Maassbestimmungen insbesondere Zuriick-
fiihrung der Stronrintensitiits - Messuugen auf
nieckanisches Maass.

Abhandlungender Kouiglich Sachsischeu Gesell-
schaft der Wissenschaften (Leipzig), 1857, Bd. 5,
pp. 219-292.

360. King (Wm. P.). Description of Sir Wm. Thom-
son's experiments made for the determination of
i', the number of electrostatic units in the electro-
magnetic unit.

Report of the 39th meeting of the British Asso-
ciation for the Advancement of Science, 1869, pp.
434-436. Also, pp. 186-188 of Reports of the Com-
mittee on Electrical Standards appointed by the
British Association for the Advancement of
Science. Reprinted by permission of the Council.
Londou : 1873, 8vo, 248 pp.

3G1. Maxwell (J. Clerk). On a method of making a
direct comparison of electrostatic with electro-
magnetic force ; with a note on the electromag-
netic theory of light.
Phil. Trans., 1868, pp. 643-657.

362. M'Kichan (Dugald). Determination of the num-

ber of electrostatic units in the electromagnetic
unit, made in the Physical Laboratory of Glasgow
University.

Phil. Trans., 1873, pp. 409-427.

363. Ayrton (W. E.) and Perry (John.) Anewdeternii-

nation of the ratio of the electro-magnetic to the
electro-static unit of electric quantity.

Journal of the Society of Telegraph Eugineers
(London), 1879, vol. 8, pp. 126-141. Also, L., E. &
D. Phil. Mag., 1879, vol. 7, pp. 277-2-'.».

364. Hockin (C). Note on the capacity of a certain

condenser, and on the value of v.

Report of the 49th meeting of the British Asso-
ciation for the Advancement of Science, 1879, pp.
285-290.

365. Shida (R.). On the number of electrostatic units

in tin; electromagnetic unit.

L., E. <fc D. Phil. Mag., 1880, vol. 10, pp. 431-
436.

366. Stoletow (A.). Sur une mdthode pour determiner

le rapport des unites electromagnetiqiies et clec-
trostatiques (le v de Maxwell). (3 figures.)

Journal de Physique (Paris), 1881, vol. 10, pp.
468-474.

367. Exner (Franz). Bestimmung des Verhiiltnisses

zwischen elektrostatischer uud elektromagneti-
scher absoluter Einheit.

Sitzungsberichte der Matkeinatisch-Naturwis-
senschaftlichen Classe der kaiserlichen Akademie
der Wissenschaften (Wien), 1882, 86. bd., II. Ab-
theilung, pp. 106-114.

368. Thomson (J. J.). On the determination of the

number of electrostatic units in- the electromag-
netic unit of electricity.

Phil., Trans. 1883, pp. 707-722.

369. Klemencic (Ignaz). Untersuchungen iiber das

Verhaltniss zwischen dem elektrostatischeu und
elektromagnetischen Maass-System. I.

Sitzungsberichte der Mathematisch-Naturwis-
senschaftlichen classe der kaiserlichen Akademie
der Wissenschaften (Wien), 1884, 89. bd., II. Ab-
theilung, pp. 298-328.

ON Mil SOLAR PARALLAX AND ITS RELATED O

1 6

370.

371

Himstedt (F.). Uober die Bestimmnng der Capa-
citiit eines Schutzring-Condensators in absolntem,
eleotro-magnetischem Maasso.

Annalen der Physik und Chemie * • * von
G. Wiedemann, Leipzig, 1888, Bd. 35, pp. 126-136.

Rowland (Henry A.)- On tbe ratio of the electro-
static to the electromagnetic unit of electricity.

L., E. & D. Phil. Ma.-.. 1889, vol. 28, pp. 304-315.
Also, Amer. Jour, of Science, 1889, vol. 38, pp.
289-2

372.

373

Rosa (Edward B.). Determination of r, the ratio
of the electromagnetic to the electrostatic unit.

L., E.<S D. Pliil. Mag., L889,T0l.28,pp.315
Alto, Amer. Jour, of Science, 1839, vol. 38, p]>.
298-312.
Thomson (J. J.) and Searle (G. F. C). A determi-
nation of" r, " the ratio of the electromagnetic unit
of electricity to the electrostatic unit.
Proc. Roy! Soc, 1890, vol. 47, pp. 376-378

\

I NDEX.

Page.

Aberration, constant of 25

adjusted value of the constant of .73, 108, 111, 133, 142

no satisfactory theory of 144

Adjusted quantities, constants necessary for find-
ing 127

sources of their probable errors 142

Airy, Sir Geo. B 4,7,8,9,12,13,14,15,17,18,34,52,

53, 89, 90, 91, 97, 113, 101, 143

Alembert, J. lo R. d' 58

Algebraic notation 1

Angle of the vertical, formula for 139

Anomalistic month, length of 16

Asten, E. von 34,35

Asteroids, solar parallax from observations of. . . 51

Authorities, mode of citing 1, 146

Auwers, A 24,25,26

Ayrton, W. E 135

Backlund, O 34

Bailie, J. B 89,90

Bailly, J. S 28

Baily,F 89,90,97

Baird, Maj. A. W 113,114,115,118,119

Ball, L. de 25

Bernoulli, D, 112

Bessel, F. W 4, 13, 23, 24, 25, 34, 35, 36, 86, 87, 93, 137

Bibliography 146

Biot, J. B 6,8

Bolto 24

Bond, G. P 35,36

Bond, W. C 51,54

Borda, J. C 6,7

Boreneus, H. G 97

Bouguer, P 3,89,90

Bouvard, A 34,35

Bradley, Jas 23,24,25,26,44,51,101

Breen, H 21

Brinkley, John 25,26

Biirg, J. T 101

Burckhardt, J. C 20,34,101

Buscb, A. L 25,26

Campbell, James 19

Carlini, F 89,90,91

Cassini, J. D 51

Cavendish, H 89,90

Page.
Centrifugal forco at the surface of the Earth .. . 56

Citation of authorities, modo of 1, 146

Clairaut, A. C 9,94,97

Clarke, Gen. A. E 3,4,5,9,89,90,91,90,97,98,

102,109,110,111,126,138

Clarke, F. W 91,92

Coast Survey, arcs projected by the United

States 143

Computations, how made 146

Comstock, Gen. C. B 143

Concluding remarks 142

Conditional equations, general forms of 56

Constants, desiderata for improvement of astro-
nomical 145

Copeland, Ralph 35

Cornu, A 29,30,33,89,90

Corrections by adjustment, numerical values of. 68, 108
Cruls, L 52,53

D'Alembert, J. le R 58

Damoiseau, M. C. T. de 13,20

Darwin, Geo. H 98,113,114.115,118,119

Day, length of sidereal 10, 139

Delambre, J. B.J 3,28,34,52

Delaunay, Ch 17,18,60,62

Density of the Earth 89,139

Desiderata for improvement of astronomical con-
stants 145

Distances of the Sun and Moon. See under Sun

and Moon.
Distances of Sun and Moon, formula for probable

error of 72,134

Diurnal tidal oscillations, Ferrel's expressions for 114

Downing, A. M. W 25,26,27,51,54

Dreyer, J. L. E 24

Du Sejour, D 52

Earth, centrifugal force at surface of 56

density, flattening, and moments of inertia

of B9

observed mean density of 89, 139

observed surface density of 91, 139

Le Gendre's law of density of interior of 95

adjusted value of eccentricity of 138

adjusted value of flattening of 110, 111, 133, 138

flattening of, found from perturbations of

the Moon

165

i66

IX HEX.

\

Tage.
Earth, etc. — Continued.

flattening of, Hansen's formula for finding,

from perturbations of the Moon 100

flattening of, Pontdcoulant's formula for

finding, from perturbations of the Moon. 100

values of the flattening of, found by various

authors, from perturbations of the Moon. 101

flattening of, deduced from precession and

nutation 92

flattening of, found by various authorities

from pendulum experiments 97,137,138

uncertainty respecting the flattening of 142

uncertainty in the flattening of, and its effect

upon the other constants 102

mass of.. 34,42,49

adjusted mass of 73,108,111,133,140

general expression for adjusted mass of 107

moments of inertia of 93,99,100,101,102, 139

table of processional moment of inertia of,

according to various authors 93

lunar inequality of motion of 19

formula for lunar inequality of motion of. .. 62
adjusted value of lunar inequality of mo-
tion of 73,108,111,133,140

eccentricity of orbit of 11, 140

perturbations of plane of orbit of 74

morion of perigee of , 45

perturbations of, by Mars 45

perturbations of, by Venus 45

formula for rail ins of 139

size and figure of 3,109,133,136, 138

quadrant, formula for length of... 134

quadrant, length of 133, 138

Eastmau, J. R 51,54

Eccentricity, of the Earth..." 138

of the Earth's orbit 11,140

of the Moon's orbit 18,140

Ecliptic, obliquity of 22, 79, 80, 80, 141

perturbations of plane of 74

Electric constant, v 135

Encke, J. F 33,34,37,52,53

Euler, Leonhard 52

Enlerian nutation, period of 145

Exner, Franz 135,136

Ferrel, Wm 112, 113, 114, 115, 110, 117, 118, 119

Ferrer, J. J.de 52

Finlayson, H. P 112

Fischer,A 97,136

Fizcau.H.L 29,33

Flamstead, John 51

Flattening of the Earth, found from pendulum
experiments, according to various authori-
ties 97,137,138

found from perturbations of the Moon 98

found from precession and mutation 92

uncertainty respecting 142

Forbes, G 30,33

Foucault, L 29,33

Frisby, Edgar 146

Galle, J.G 51,54,55

Geodetic formula) 3,5,104,109,134

Page.

Gill, David 51,54,55

Gilliss, Jas. M 51,54

Giulio, Chas.1 89,90

Glasenapp, S 28

Gould, B.A 51,54

Gravity, acceleration by 133, 139

Gyld6n, J.A.H 26,27

Haerdtl, E. von 34, 35

Hall, Asaph 2G, 27, 34, 35, 36, 51, 54, 146

Sail, Asaph, jr 35,36

Hall, Maxwell 51,54

Hansen, P. A. ..10, 11, 12, 13, 15, 10, 17, 18, 19, 34, 37, 38, 61,
62, 78, 80, 86, 87, 93, 98, 100, 101, 102, 142

Harkness, Wm 53,91

Harmonic analysis of the tides, notation used by

various authors in the 114

Haughton, S 89,112

Helmert, F. R 8, 9, 97, 102, 110. 126, 142, 143

Henderson, Thos 20, 21 , 26, 51, 54

Herschel, Sir F. W 35

Hill, Geo. W 34,37,3^,39,41,97,98,100,102,130

Himstedt, F 135

Hind, J. R 35

Hockin, C 135

Holden, E. S 35,36

Bornsby, Thomas 52

Horrox, Jeremiah 44

llnttou, Charles 89,90

Inclination of the Moon's orbit 18,140

Indian arc, effect of, on computed length of the

Earth's polar semidiameter 143

Introduction 1

James, Sir Henry 89,90,91

Jolly, Ph. von 89, '.10

Jupiter, mass of 34,35,42, 140

Kilometer, length of 3

King, W. F 135,136

KlemenSic, Ignaz 135

Kohlrausch, R. H. A 135, 136

Krueger, A. 34,35

Kiistner, F 26,27

Lacaille, N. L. de 20,51

La Condamine, C. M. de 3,89,90

LaGrange, J. L 33,34,35

Lake Survey, arcs measured by the United States 143

Lalande, J. J. le F. de 24,52

Lamont, J 35

La Place. P. S. de 6,8,34,35,86,87,93,94,98,

99, 101, 112. 113

Lassell, W 35,36

Latitude, period of variation of 145

Least square adjustment, general formula for

the 64,106,122

a more comprehensive 121

LeGendre, A. M 92,94,95,96,98,102,143

LeGeudrc's law of density of the interior of tho

Earth 95

INDEX.

167

Tago.

Length, relations between standards of 3

Le Verrier, U. J ... 10, 1 1, 12, 19, 22, 34, 35, 36, 40, 41, 42, 44,
45, 73, 74, 75, 76, 78, 79, 80, 86, 87, 93, 94

L6vy, M 98

Lexell, A.J 52

Light equation, the 28

adjusted value of 73, 108, 111, 133, 14-2

Light, observed velocity of 29

adjusted value of velocity of 73, 108. 111,133, 142

Lindenau, B. von 26,27,31

Lindhagen, D. G 26,27

Lindsay, Lord 51,54

Lipschitz, K 98

Lubbock, Sir J. W 112

Lunar inequality of Earth's motion 19

inequality of the Earth's motion, adjusted
value of 73,108,111,133,140

parallax 20

parallax, adjusted value of 73,108,111, 133, 140

Luni-solar precession. Sec under Precession.

Lundahl, G 25,26,27

Maclear, Sir Thomas 26.27

Main, Robert 25,26,27

Mars, mass of 31,35, 42, 140

motions of its perihelion, eccentricity, incli-
nation and node 43

position of, in 1672 44

solar parallax from observations of 51

Marth, A 35,36

Maskelyne, Nevil 28,52,89,90,101

Mass of Moon, formula for. 60,105,122

(See Moon.)

Masses of the planets 33,42,49, 140

(See under name of each planet.)
formula' for changes in, on account of cor-
rections by adjustment 68, 107

formulae for probable error of 49,107

Mathieu, C. L 6,8

Maxwell, James Clerk 135, 136

M'Kichan, Dugald 135,136

Mean density of the Earth 89,139

M.xbain, P. F. A 3

Mendenhall, T. C 89,90

Mercury, longitude of node of 36

mass of 33,42,49

general expression for adjusted mass of 107

adjusted mass of 73,108,111,133,140

motion of node of 40

Meter, length of 3

Michelson, A. A 29,30,33

Midler, A 34,35

Moments of inertia of the Earth. ..93,99, 100, 101, 102, 139

Mouth, length of 12,139

anomalistic 16

nodical - 16

sidereal 15, 139

synodical 16, 139

tropical 16

Moon, distance of 73, 108, 110, 111, 133, 140

formula for probable error of distance of. . . . 72, 134

eccentricity of orbit of 18, 140

inclination of orbit of 18, 140

adjusted mass of 73,108, 111, 133, 140

mass of, from observations of the tides.. 112, 118, 121

Moon, etc. — Continued.

expressions for computing mass of, from

diurnal tidal osoillationa 115, 116, 117

expressions for computing massof, from semi-
diurnal tidal oscillations 115, 116, 117

n 1 can sidi nal motion of 13

motion of node of 16,140

parallactic inequality of in

formula for parallactic inequality of 00

adjusted value of parallactic inequality of. . . 7;:, 108,

111,1:::!, Ill

parallax of 20

adjusted value of parallax of 73,108,111,133,140

Neison, E 19

Neptune, mass of 36, 42, 140

Newcomb, Simon. .12, 13, 14,15,16, 17 I- !3,30,33,

34,35,36,51,54,61,62, 138

Newton, Sir Isaac 34, 35, 112

Nicolai, F. B. G 34

Node, motion of Moon's 16, 140

Nodical mouth, length of 16

Notation, key to algebraic 1

Notation of various authors for harmonic anal-
ysis of the tides 114

Nutation, constant of 25

formulae for adjusted values of 79,141

formula for, in terms of the precession and

lunar parallax 60, 105, 122

Dr. Peters's formula for 87

adjusted value of 73, 108, 111, 133, 141

period of Etilerian 145

Nyr6n, Magnus 23,24,25,26,27,137,138

Obliquity of ecliptic 22,79,80,86,141

Obrecht, A 53,54

Observed quantities, values of 68,106,128,133

Olufsen, C. F. R 10,11,20,21,34,37,38

Parallactic inequality of the Moon's motion 18

Parallactic inequality of the Moon, adjusted

value of 73, 108, 111, 133, 141

Parallax. See Solar Parallax, and Lunar parallax.

Paris line, length of 3

Pauckcr, M. G. von 97

Pechmann, E 89,90

Peirce, Benjamin 35, 120

Pendulum, formuhe relating to. .6, 7, 9, 56, 97, 103, 134, 137

length of the seconds 6, 133, 136

ad j usted length of tho seconds 1 39

Perry, John 135

Perturbations. See under Mercury, Venus,
Earth, and Mars.

Peters, C. A. F 25,26,27,86,87,93,137

Piazzi, G 23,25,26

Picard, J 41

Pingre", A. G 52

Plana, Jean 34,91,98

Planets, masses of 33, 42, 49, 140

Planman, A 52

Playfair, John 89,90,91

Poiucare", H 9?

Poisson, S. D "• 8

Pout6coulant, G. de 10°

1 68

INDEX.

Page.

Pound, James 51

Powalky, C 34.52,53

Poynting.J. H 89,90

Pratt, John H 136

Precession, constant of 22

formula for changes in the luui-solar, on ac-
count of corrections hy adjustment 69

additional formula for 73

algebraic formula for changing the funda-
mental equinox and ecliptic 80,85

adjusted value of 73,108,111,133,141

and nutation, numerical formula; for 79,141

complete numerical formula for, applicable

to any equinox or ecliptic 86, 141

table of values of, given by various astron-
omers - 87

Probable errors, of observed quantities, corrected

values of 70,132

of the adjusted quantities, constants neces-
sary for finding 127

of adjusted quantities, sources of 142

of planetary masses, method of computing. 49, 107

Puiseux, V. A 52,53

Reich, F 89,90

Results, summary of 13-

Richardson, W 26

Richer, J 44

Roberts, Edward 113,114

Robinson, T. R 25

Roche, E 98

Roemer, O 44

Rosa, E. B 135

Rosse, Earl of 35

Rothman.R. W 34

Rowland, II. A 135

Sabine, Gen. Sir Edwai d 6, 8, 9

Safford, T. H 35

Saigey, J. F 7 8,9,89,90

Saturn, mass of 35, 42. 140

Schjellerup, H. C. F. C 23,24

Schubert, Gen. T. F. de 4

Schur.W 34,35

Schweizer, K. G 26

Searle, G. F. C 135

Seconds pendulum, length of the 6,133,136

adjusted length of the 139

Semi-diurnal tidal oscillations, Ferrel's expres-
sions for 115

Serret, J. A 22,59,77,79,92,93

Shida, R 135

Short, James 52

Sidereal day, length of 10, 139

month, length of 15

year, length of 10

Solar parallax, formula for, in terms of the Earth's

mass 57, 58, 105, 122

formula for, in terms of P, L, E and M 63, 122

formula for, in terms of P, Q, E and M 61.1 22

formula for, in terms of V and a 63, 122

Page.
Solar parallax, etc. — Continued.

formula for, in terms of V andO 63, 122

trigonometrical determinations of 51,54

from observations of Asteroids 51

from observations of Mars 51

from photographic observations . 53

from transits of Venus 51,138

adjusted value of 73,108,111,133,140

Standards of length, relations between 3

Sterneck, R. von 89,90,91

Stockwell, John N 88

Stokes, Sir Geo. G 100

Stone, E. J 19, 21, 25, 51, 52, 53, 54, 58, 59

Struve, F. G. W 25,26,27

Struve, H 35

Struve, Ludwig 24

Struve, Otto 23,24,35,86,87

Sun and Moon, ratio of mean sidereal motions of. 17, 139

Sun, distance of 73, 108, 110, 111, 133, 140

Sun, parallax of. See Solar parallax.

formula for probable error of distance of 72, 134

adjusted value of parallax of 73, 103, 111, 133, 140

equation of center 45

Summary of results 138

Supplementary data 136

Surface density of the Earth 91, 139

Sy nodical month, length of 16,139

Tait, P. G 94

Taylor, T. G 51,54

Thomson, J. J 135

Thomson, Sir Wm 94, 113, 135, 136

Tidal oscillations, Ferrel's expressions for the

diurnal 114

Ferrel's expressions for the semi-diurnal 115

Tidal observations at thirty-four different sta-
tions, Moon's mass computed from 118, 121

Tides, notation used by various authors in the

harmonic analysis of 114

Tisserand, F 34,44,98

Todd, D. P 53,54

Toise, length of 3

Transits of Venus, solar parallax from 51,138

Transit of Venus Commission, United States 38,54

Tropical month, length of 16

Tropical year, length of 10

Tupman, G. L 52,53

Unferdinger, F 7,8,9,97,143

Units of length, relations between 3

Uranus, mass of 35,42, 140

v, the electric constant 135

Velocity of light, observed value of 29

adjusted value of 73,108,111,133,142

Venus, longitudes of 41

mass of 34,42. 19

adjusted mass of 73,108,111,133,140

general expression for adjusted mass of 107

longitude of node of 37

motion of node of 40

Vertical, formula for angle of 139

INDI-A.

I 69

Page.

Wagner, A 2(>/27

Weber, W. E 135, 13C

Weisse, Maximilian 23

Wilsing, J 89,90

Winnecke, A 51,54

Woodward, R.&..J. 14G

6987 12

Warm, J.P 34,36

Year, length of the sidereal 1". 139

length of the tropical 10, I ■"■'

Young, James 30

Zi

I